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* BY 


M.lNST.C.E., FEL.A,L!E., M.I.E.E. 

Professor of Electrical Design, Purdue University, 

Author of " Polyphase Currents," 
"Electric Power Transmission," etc. 















A BOOK which deals exclusively with the theory 
and design of alternating current transformers is not 
likely to meet the requirements of a College text to 
the same extent as if its scope were broadened to include 
other types of electrical machinery. On the other 
hand, the fact that there may be a limited demand for 
it by college students taking advanced courses in elec- 
trical engineering has led the writer to follow the 
method of presentation which he has found successful 
in teaching electrical design to senior students in the 
school of Electrical Engineering at Purdue University. 
Stress is laid on the fundamental principles of electrical 
engineering, and an attempt is made to explain the 
reasons underlying all statements and formulas, even 
when this involves the introduction of additional 
material which might be omitted if the needs of the 
practical designer were alone to be considered. 

A large portion of Chapter II has already appeared 
in the form of articles contributed by the writer to the 
Electrical World; but the greater part of the material 
in this book has not previously appeared in print. 

January, 1919 









1. Introductory i 

2. Elementary Theory of Transformer 2 

3. Effect of Closing the Secondary Circuit 6 

4. Vector Diagrams of Loaded Transformer without Leakage. ... 10 

5. Polyphase Transformers 12 

6. Problems of Design 13 

7. Classification of Alternating-current Transformers 14 

8. Types of Transformers. Construction 17 

9. Mechanical Stresses in Transformers 24 



10. The Dielectric Circuit 32 

11. Capacity of Plate Condenser 40 

12. Capacities in Series 42 

13. Surface Leakage 46 

14. Practical Rules Applicable to the Insulation of High-voltage 

Transformers 48 

15. Winding Space Factor 51 

16. Oil insulation 52 

17. Terminals and Bushings , 54 

18. Oil-filled Bushing 57 

19. Condenser-type Bushing 62 






20. Losses in Core and Windings 69 

21. Efficiency % 73 

22. Temperature of Transformer Windings 79 

23. Heat Conductivity of Insulating Materials 80 

24. Cooling Transformers by Air Blast 88 

25. Oil-immersed Transformers, Self-cooling 91 

26. Effect of Corrugations in Vertical Sides of Containing Tank ... 94 

27. Effect of Overloads on Transformer Temperatures 98 

28. Self-cooling Transformers for Large Outputs 103 

29. Water-cooled Transformers 105 

30. Transformers Cooled by Forced Oil Circulation 106 


31. Magnetic Leakage 107 

32. Effect of Magnetic Leakage on Voltage Regulation 109 

33. Experimental Determination of the Leakage Reactance of a 

Transformer 114 

34. Calculation of Reactive Voltage Drop 117 

35. Calculation of Exciting Current 1 25 

36. Vector Diagram Showing Effect of Magnetic Leakage on 

Voltage Regulation of Transformers 132 



37. The Output Equation 138 

38. Specifications 140 

39. Estimate of Number of Turns in Windings 141 

40. Procedure to Determine Dimensions of a New Design 149 

41. Space Factors : 151 

42. Weight and Cost of Transformers 151 

43. Numerical Example 154 





44. General Remarks J 77 

45. Transformers for Large Currents and Low Voltages i?7 

46. Constant Current Transformers i?8 

47. Current Transformers for use with Measuring Instruments 183 

48. Auto-transformers JQ 1 

49. Induction Regulators *97 


,4= area of equipotential surface perpendicular to lines of force 

(sq. cm.). 

A cross-section of iron in plane perpendicular to laminations (sq. in.). 
a = ampere- turns per inch length of magnetic path. 
a = total thickness of copper per inch of coil measured perpendicularly 

to layers. 

B = magnetic flux per sq. cm. (gauss). 
Bam is defined in Art. 9. 

b = total thickness of copper per inch of coil measured through insu- 
lation parallel with layers. 

C electrostatic capacity; or permittance, 

coulombs _ r ,r IN 

= =nux per unit e.m.f. (farad). 

Cmf = capacity in microfarads. 

c=a coefficient used in determining Vt. 


D = flux density in electrostatic field =* -j = KkG (coulombs per sq. cm.). 


E= e.m.f. (volts), usually r.m.s. value, but sometimes used for max. 

1= virtual value of induced volts in primary ( =E 2 Xjr] . 

E\ = component of impressed voltage to balance E\. 
EZ = secondary e.m.f. produced by flux <; induced secondary e.m.f. 
E e primary voltage equivalent to secondary terminal voltage 




E p e.m.t. (volts) applied at primary terminals.. 
E s = secondary terminal voltage. 

-E z =irhpressed primary voltage when secondary is short-circuited. 
e=e.m.f. (volts). 

F= force (dynes). 

/= frequency (cycles per second). 

G=^ = potential gradient (volts per centimeter). 

g= distance between copper of adjacent primary and secondary 
coils, in centimeters (Fig. 42). 

H= magnetizing force, or m.m.f. per cm. 
h= length (cms.) defined in text (Fig. 42). 

7 = r.m.s. value of current (amps.). 

/i = balancing component of primary current = I s ( ) . 

I c = current in the portion of an auto-transformer winding common 

to both primary and secondary circuits. 
Ie = total primary exciting current. 

/o = " wattless " component of I e (magnetizing component). 
Ip = total primary current. 
/o = total secondary current. 
/; = " energy " component of I e (" in-phase " component). 

J = 8.84Xio- 14 farads per cm. cube = the specific capacity of air. 

K 1 
' > definition follows formula (34) in Art. 27. 

k dielectric constant or relative specific capacity, or permittivity 

(k = i for air). 

& = heat conductivity (watts per inch cube per i c C.). 
k= coefficient used in calculating the effective cooling surface of 

corrugated tanks. 
k c = about i.SXio- 6 for copper. 
ki= (refer text (Art. 39) for definition). 

1= length (cms.). 

/=mean length, in centimeters, of projecting end of transformer c,oil. 

1 = length measured along line or tube of induction (cms.). 


; c = mean length per turn of windings. 

/i = mean length of magnetic circuit measured along flux lines. 

M c = weight of copper in transformer coils (Ibs.) 
if o = weight of oil in transformer tank (Ibs.). 

W=27T/XlO~ 8 . 

n = - (in formula for calculating cooling surface of corrugated tanks). 


= usually from 1.6 to 2 in B H (core loss formulas). 

P = weight of iron in transformer core (or portion of core), Ibs. 

p = thickness of half primary coil in centimeters (denned in text in 
connection with Fig. 42). 

R = resistance (ohms). 
Ri = resistance of primary winding (ohms). 
R 2 = resistance of secondary winding (ohms). 

Rh = " thermal ohms." 

IT P \ z 
R p equivalent primary resistance = Ri+R 2 [=- 

r = ratio ^ J-T T (auto-transformers). 

number of turns common to both circuits 

S = effective cooling surface of transformer tank (sq. in). 

5 = thickness of half secondary coil (cms.) denned in text (Fig. 42). 

T = number of turns in coil of wire. 

Ti = number of turns in half primary group of coils adjacent to 

secondary coil. 
To = number of turns in half secondary group of coils adjacent to 

primary coil. 

Td = difference of temperature (degrees centigrade). 
To = initial oil temperature. 
T p = number of turns in primary winding. 
T s = number of turns in secondary winding. 
7\ = oil temperature at end of time t m minutes. 
/ = thickness (usually inches). 
t = interval of time (seconds). 
t m = interval of time (minutes). 

Vt = volts induced per turn of transformer winding. 


W = power (watts). 
W c = full-load copper loss (watts). 
Wt = core loss (watts). 
Wt = total transformer losses (watts). 

w = watts dissipated per sq. in. of (effective) tank surface. 

w = watts lost per Ib. of iron in (laminated) core. 

Xi = reactance (ohms) of one high-low section of winding. 
Xp = reactance (ohms) commonly referred to as equivalent primary 

ZD = impedance (ohms) on short circuit. 

= phase angle (cos 6 = power factor of external circuit). 
e = " electrical " angle (radians) = 2w ft. 
X = pitch of corrugations on tank surface. 
* = magnetic flux (Maxwells) in iron core. 

= phase angle (cos = power factor on primary side of transformer). 
SF = dielectric flux, or quantity of electricity, or electrostatic induc- 
tion = CE = A D coulombs. 





1. Introductory. The design of a small lighting 
transformer for use on circuits up to 2200 volts, or 
even 6600 volts, is a very simple matter. The items 
of importance to the designer are: 

(1) The iron and copper losses; efficiency, and tem- 
perature rise; 

(2) The voltage regulation, which depends mainly 
upon the magnetic leakage, and therefore upon the 
arrangement of the primary and secondary coils; 

(3) Economical considerations, including manufac- 
turing cost. 

With the higher voltages and larger units, not only 
does the question of adequate cooling become of greater 
importance; but other factors are introduced which 
call for considerable knowledge and skill on the part 
of the designer. The problems of insulation and pro- 


tection against abnormal high-frequency surges in the 
external circuit are perhaps the most important; but 
with the increasing amount of power dealt with by 
some modern units, the mechanical forces exerted by 
the magnetic flux on short-circuits, or heavy over 
loads, may be enormous, requiring special means of 
clamping or bracing the coils, to prevent deformation 
and damage to insulation. 

Since we are concerned mainly with a study of the 
transformer from the view point of the designer, little 
will be said concerning the operation of transformers, 
or the advantages and disadvantages of the different 
methods of connecting the units on polyphase systems. 
It will, however, be necessary to discuss the theory 
underlying the action of all static transformers, and 
it is proposed to take up the various aspects of the 
subject in the following order : 

Elementary theory, omitting all considerations likely 
to obscure the fundamental principles; brief descrip- 
tion of leading types and methods of manufacture; 
problems connected with insulation; losses, heating, 
and efficiency; advanced theory, including study of 
magnetic leakage and voltage regulation; procedure 
in design; numerical examples of design; reference to 
special types of transformers. 

2. Elementary Theory of Transformer. A single- 
phase alternating current transformer consists essen- 
tially of a core of laminated iron upon which are wound 
two distinct sets of coils, known as the primary and 
secondary windings, respectively, all as shown dia- 
grammatically in Fig. i. 


When an alternating e.m.f. of E v volts is applied 
to the terminals of the primary (P), this will set up a 
certain flux (<J>) of alternating magnetism in the iron 
core, and this flux will, in turn, induce a counter e.m.f. 
of self-induction in the primary winding; the action 
being similar to what occurs in any highly inductive 
coil or winding. Moreover, since the secondary coils 

,1= 7olta 

Path of flux:J> maxwells 
linking with both windings. 

FIG. i. Essential Parts of Single-phase Transformer. 

although not in electrical connection with the pri- 
mary are wound on the same iron core, the variations 
of magnetic flux which induce the counter e.m.f. in 
the primary coils will, at the same time, generate an 
e.m.f. in the secondary winding. 

The path of the magnetic lines is usually through 
a closed iron circuit of low reluctance, in order that 


the exciting ampere- turns shall be small. There will 
always be some flux set up by the primary which does 
not link with the secondary, but the amount of this 
leakage flux is usually very small, and in any case 
it is proposed to ignore it entirely in this preliminary 
study. In this connection it may be pointed out that 
the design indicated in Fig. i, with a large space for 
leakage flux between the primary and secondary coils, 
would be unsatisfactory in practice; but the assump- 
tion will now be made that the whole of the flux 
($ maxwells) which passes through the primary coils, 
links also with all the secondary coils. In other words, 
the e.m.f. induced in the winding per turn of wire will 
he the same in the secondary as in the primary 
coils. . 

Suppose, in the first place, that the two ends of 
the primary winding are connected to constant pres- 
sure mains, and that no current is taken from the 
secondary terminals. The total flux of <i> maxwells 
increases twice from zero to its maximum value, and 
decreases twice from its maximum to zero value, in 
the time of one complete period. The flux cut per 
second is therefore 4$/, and the average value of the 
induced e.m.f. in the primary is, 


-'-'average Tr >8 OilS, 

where T p stands for the number of turns in the primary 

If we assume the flux variations to be sinusoidal, the 


form factor is i.n, and the virtual value of the induced 
primary volts will be, 

I0 8 

The vector diagram corresponding to these condi- 
tions has been drawn in Fig. 2. Here OB represents the 
phase of the flux which is set up by the current OI e in 


FIG. 2. Vector Diagram o'f Unloaded Transformer. 

the primary. This total primary exciting current can 
be thought of as consisting of two components: the 
" wattless " component 01 'o which is the true magnetiz- 
ing current, in phase with the flux; and OI W (which owes 
its existence to hysteresis and eddy current losses) 
exactly 90 in advance of the flux. The volts induced 
in the primary are OE\ drawn 90 behind OB to repre- 
sent the lag of a quarter period. The voltage that must 
be impressed at the terminals of the primary is OE P 
made up of the component OE\ exactly equal but 
opposite to OEi, and E' \E V drawn parallel to 01 e and 


representing the IR drop in the primary circuit. The 
actual magnitude of this component would be I e Ri 
where R\ is the ohmic resistance of the primary; but in 
practice this ohmic drop is usually so small as to be 
negligible, and the impressed voltage E p is virtually the 
same as E'i, i.e., equal in amount, but opposite in phase 
to the induced voltage E\. 

For preliminary calculations it is, therefore, usually 
permissible to substitute the terminal voltage for the 
induced voltage, and write for formula (i) 

> (approximately). . . (10) 

E.s= J ~~- - (approximately), . . (ib) 

where E s and T s stand respectively for the secondary 
terminal voltage and the number of turns in secondary. 
It follows that, 

1? T 


E 9 T p 

E s T s ' 

which is approximately true in all well-designed static 
transformers when no current, or only a very small 
current, is taken from the secondary. 

3. Effect of Closing the Secondary Circuit. When 
considering the action of a transformer with loaded 
secondary, that is to say, with current taken from the 
secondary terminals, it is necessary to bear in mind that 
except for the small voltage drop due to ohmic resist- 
ance of the primary winding the counter e.m.f. induced 


by the alternating magnetic flux in the core must still 
be such as to balance the e.m.f. impressed at primary 
terminals. It follows that, with constant line voltage, 
the flux <f> has very -nearly the same value at full load as 
at no load. The m.m.f. due to the current in the sec- 
ondary windings would entirely alter the magnetization 
of the core if it were not immediately counteracted by 
a current component in the primary windings of exactly 
the same magnetizing effect, but tending at every instant 
to set up flux in the opposite .direction. Thus, in order 
to maintain the flux necessary to produce the required 
counter e.m.f. in the primary, any tendency on the part 
of the secondary current to alter this flux is met by a 
flow of current in the primary circuit; and since, in' 
well-designed transformers, the magnetizing current is 
always a small percentage of the full-load current, it 
follows that the relation 

I.T^LT,, (3) 

is approximately correct. 

Thus, !, 

is J- p 

where I p and I s stand respectively for the total primary 
and secondary current. 

The open-circuit conditions are represented in Fig. 3 
where E v is the curve of primary impressed e.m.f. and 
L is the magnetizing current, distorted by the hysteresis 
of the iron core, as will be explained later. E s is the 


curve of secondary e.m.f. which coincides in phase with 
the primary induced e.m.f. and is therefore if we 
neglect the small voltage drop due to ohmic resistance 
of the primary exactly in opposition to the impressed 
e.m.f. The curve of magnetization (not shown) would 

FIG. 3. Voltage and Current Curves of Transformer with Open Second- 
ary Circuit. 

be exactly a quarter period in advance of the induced, 
or secondary, e.m.f. 

In Fig. 4, the secondary circuit is supposed to be closed 
on a non-inductive load, and the secondary current, 
I s will, therefore, be in phase with the secondary 


The tendency of the secondary current being to pro- 
duce a change in the magnetization of the core, the cur- 
rent in the primary will immediately adjust itself so as 
to maintain the same (or nearly the same) cycle of mag- 
netization as on open circuit; that is to say, the flux 

FIG. 4. Voltage and Current Curves of Transformer on Non-inductive 


will continue to be such as will produce an e.m.f. in the 
primary windings equal, but opposite, to the primary 
impressed potential cjifference. The new curve of pri- 
mary current, I v (Fig. 4), is therefore obtained by adding 
the ordinates of the current curve of Fig. 3 to those of 
another curve exactly opposite in phase to the secondary 



current, and of such a value as to produce an equal mag- 
netizing effect. 

4. Vector Diagrams of Loaded Transformer Without 
Leakage. The diagram of a transformer with secondary 
closed on a non-inductive load is shown in Fig. 5. In 
order to have a diagram of the simplest kind, not only 
the leakage flux, but also the resistance of the windings 

El I> , I u I. E 2 

FIG. 5. Vector Diagram of Transformer on Non-inductive Load. 

will be considered negligible. The vectois then have the 
following meaning: 

QB = Phase of flux $ linked with both primary and 

secondary windings; 

I e = Exciting current necessary to produce flux 3>; 
Ez Secondary e.m.f. produced by alternations of 

the flux $; 
E'i= Primary e.m.f. equal, but opposite, to the 

e.m.f. produced by alternations of the flux $ 

(In this case it is equal to the applied e.m.f., 

since the IR drop is negligible) ; 


I s = Current drawn from secondary ; in phase with 2 ; 
/i = Balancing component of primary current, drawn 


exactly opposite to I s and of value / s X^r; 

IP = Total primary current, obtained by combining 

/i with I e . 

In Fig. 6 the vectors have the same meaning as above, 
but the load is supposed to be partly inductive, which 
accounts for the lag of I s behind 2- 

FIG. 6. Vector Diagram of Transformer on Inductive Load. 

It is convenient in vector diagrams representing both 
primary and secondary quantities to assume a i : i 
ratio in order that balancing vectors may be drawn 
of equal length. The voltage vectors may, if preferred, 
be considered as wits per turn, while the secondary 
current vector can be expressed in terms of the pri- 
mary current by multiplying the quantity representing 


the actual secondary current by the ratio ~. 

L v 


5. Polyphase Transformers. Although we have con- 
sidered only the single-phase transformer, all that has 
been said applies also to the polyphase transformer 
because each limb can be considered separately and 
treated as if it were an independent single-phase trans- 

In practice it is not unusual to use single-phase trans- 
formers on polyphase systems, especially when the units 
are of very large size. Thus, in the case of a three-phase 
transmission, suppose it is desired to step up from 6600 
volts to 100,000 volts, three separate single-phase trans- 
formers can be used, with windings grouped either Y 
or A, and the grouping on the secondary side need not 
necessarily be the same as on the primary side. A 
saving in weight and first cost may be effected by com- 
bining the magnetic circuits of the three transformers 
into one. There would then be three laminated cores 
each wound with primary and secondary coils and joined 
together magnetically by suitable laminated yokes ; 
but since each core can act as a return circuit for the 
flux in the other two cores, a saving in the total weight 
of iron can be effected. Except for the material in the 
yokes, this saving is similar to the saving of copper in 
a three-phase transmission line using three conductors 
only (as usual) instead of dx, as would be necessary 
if the three single-phase circuits were kept separate. 
In the case of a two-phase transformer, the windings 
would be on two limbs, and the common limb for the 
return flux need only be of sufficient section to carry 
\/2 times the flux in any one of the wound limbs. 

It is not always desirable to effect a saving in first 


cost by installing polyphase tiansformers in place of 
single-phase units, especially in the large sizes, because, 
apart from the increased weight and difficulty in hand- 
ling the polyphase transformer, the use .of single-phase, 
units sometimes leads to a saving in the cost of spares to 
be carried in connection with an important power devel- 
opment. It is unusual for all the circuits of a polyphase 
system to break down simultaneously, and one spare 
single-phase transformer might be sufficient to prevent 
a serious stoppage, while the repair of a large polyphase 
transformer is necessarily a big undertaking. 

6. Problems in Design. The volt-ampere input of a 
single-phase transformer is E P I P , and if we substitute 
for E p the value given by formula (ia), we have 

A A.A.f 

Volt-amperes = -~ X $ X TJ P . 

Thus, for a given flux <, which will determine the cross- 
section of the iron core, there is a definite number of 
ampere turns which will determine the cross-section of 
the winding space. There is no limit to the number of 
designs which will satisfy the requirements apart from 
questions of heating and efficiency; but there is obvi- 
ously a relation between the weight of iron and weight 
of copper which wjll produce the most economical 
design, and this point will be taken up when discussing 
procedure in design. It will, however, be necessary to 
consider, in the first place, a few practical points in 
connection with the construction o;' transformers, and 
also the effect of insulation on the space available for the 
copper. The predetermination of the losses in both iron 


and copper must then be studied with a view to calcu- 
lating the temperature rise and efficiency. Finally, the 
flux leakage must be determined with a reasonable 
degree of accuracy because this, together with the ohmic 
resistance of the windings, will influence the voltage 
regulation, which must usually be kept within specified 

7. Classification of Alternating-current Transformers. 
Since we are mainly concerned with so-called constant- 
potential transformers as used on power and lighting 
circuits, we shall not at present consider constant-current 
transformers as used on series lighting systems and in 
connection with current-measuring instruments; neither 
shall we discuss in this place the various modifications 
of the normal type of transformer which render it avail- 
able for many special purposes. 

Transformers might be classified according to the 
method of cooling, or according to the voltage at the 
terminals, or, again, according to the number of phases 
of the system on which they will have to operate. 

Methods of cooling will be referred to again later when 
treating of losses and temperature rise; but, briefly 
stated, they include: 

(1) Natural cooling by air. 

(2) Self-cooling by oil; whereby, the natural circula- 
tion of the oil in which the transformer is immersed car- 
ries the heat to the sides of the containing tank. 

(3) Cooling by water circulation: a method generally 
similar to (2) except that coils of pipe carrying running 
water are placed near the top of the tank below the 
surface of the oil. 


(4) Cooling with forced circulation of oil: a method 
used sometimes when cooling water is not available. It 
permits of the oil being passed through external pipe 
coils having a considerable heat-radiating surface. 

(5) Cooling by air blast; whereby a continuous stream 
of cold air is passed over the heated surfaces, exactly 
as in the case of large turbo-generators. 

In regard to difference of voltage, this is mainly a 
matter of insulation, which will be taken up in Chap. 
II. The essential features of a potential transformer 
are the same whether the potential difference at ter- 
minals is large or small, but the high-pressure trans- 
former will necessarily occupy considerably more space 
than a low-pressure transformer of the same k.v.a. 
output. The difficulties of avoiding excessive flux leak- 
age and consequent bad voltage regulation are increased 
with the higher voltages. 

Low-voltage transformers are used for welding metals 
and for any purpose where very large currents are nec- 
essary, as for instance, in thawing out frozen water 
pipes, while transformers for the highest pressures are 
used for testing insulation. Testing-transformers to give 
up to 500,000 volts at secondary terminals are not 
uncommon, while one transformer (at the Panama- 
Pacific Exposition of 1915) was designed for an output 
of 1000 k.v.a. at 1,000,000 volts. This transformer 
weighed 32,000 lb., and 225 bbl. of oil were required to 
fill the tank in which it was immersed. 

A classification of transformers by the number of 
phases would practically resolve itself so far as present- 
day tendencies are concerned into a division between 


single-phase and three-phase transformers. From the 
point of view of the designer, it will be better to consider 
the use to which the transformer whether single-phase 
or polyphase will be put. This leads to the two 

(1) Power transformers. 

(2) Distributing transformers. 

Power Transformers. This term is here used to 
include all transformers of large size as used in central 
generating stations and sub-stations for transforming 
the voltage at each end of a power transmission line. 
They may be designed for maximum efficiency at full 
load, because they are usually arranged in banks, and 
can be thrown in parallel with other units or discon- 
nected at will. Artificially cooled transformers of the 
air-blast type are easily built in single units for outputs 
of 3000 k.v.a. single-phase and 6000 k.v.a. three-phase; 
but the terminal pressure of these transformers rarely 
exceeds 33,000 volts. A three-phase unit of the air- 
blast type with 14,000 volts on the high-tension wind- 
ings has actually been built for an output of 20,000 
k.v.a. For higher voltages the oil insulation is used, 
generally with water cooling-pipes. These transformers 
have been built three-phase up to 10,000 k.v.a. output 
from a single unit, for use on transmission systems up 
to 150,000 volts.* With the modern demand for larger 

*The 10,000 k.v.a. three-phase, 6600 to no,ooo-volt units in the 
power houses of the Tennessee Power Company on the Ocoee River weigh 
about 200,000 lb.; they are 19 ft. high, and occupy a floor space 20 ft. 
by 8 ft. 

Single-phase, oil-insulated, water-cooled transformers for a frequency 
of 60 cycles and a ratio of 13,200 to 150^000 volts have been built for an 
output of 14,000 k.v.a. from a single unit. 


transformers to operate out of doors, power transformers 
of the oil-immersed self-cooling type (without water 
coils) are now being constructed in increasing number. 
A self-cooling 25-cycle transformer for 8000 k.v.a. out- 
put has actually been built: a number of special tube- 
type radiators connected by pipes to the main oil tank 
are provided; the total cooling surface in contact with 
the air being about 7000 sq. ft. 

Distributing T"insformers. These are always of the 
self-cooling type, and almost invariably oil-immersed. 
They include the smaller sizes for outputs of i to 3 k.w. 
such as are commonly mounted on pole tops. These 
transformers are rarely wound for pressures exceeding 
13,000 volts, the most common primary voltage being 

In the design of distributing transformers, it is neces- 
sary to bear in mind that since they are continuously 
on the circuit, the " all-day ' : losses which consist 
largely of hysteresis and eddy-current losses in the iron- 
must be kept as small as possible. In other words, it is 
not always desirable to have the highest efficiency at 
full load. 

8. Types of Transformers. Construction. All trans- 
formers consist of a magnetic circuit of laminated iron 
with which the electric circuits (primary and secondary) 
are linked. A distinction is usually made between core- 
type and 'shell-type transformers. Single-phase trans- 
formers of the core- and shell-types are illustrated by 
Figs. 7 and 8, respectively. The former shows a closed 
laminated iron circuit two limbs of which carry the wind- 
ings. Each limb is wound with both primary and 


secondary circuits in order to reduce the magnetic leak- 
age which would otherwise be excessive. The coils may 
be cylindrical in form and placed one inside the other 
with the necessary insulation between them, or the wind- 
ings may be " sandwiched," in which case flat rect- 
angular or circular coils, alternately primary and sec- 

FIG. 7. Core-type Transformer. FIG. 8. Shell-type Transformer. 

ondary, are stacked one above the other with the requi- 
site insulation between. 

Fig. 8 shows a single set of windings on a central 
laminated core which divides after passing through the 
coils and forms what may be thought of as a shell of 
iron around the copper. The manner in which the core 
is usually built up in a large shell-type transformer is 
shown in Fig. 9. The thickness of the laminations 


varies between 0.012 and 0.018 in., the thicker plates 
being permissible when the frequency is low. A very 
usual thickness for transformers working on 25- and bo- 
cycle circuits is 0.014 in. The arrangement of the 
stampings is reversed in every layer in order to cover 
the joints and so reduce the magnetizing component 
of the primary current. A very thin coating of varnish 

FIG. 9. Method of Assembling Stampings in Shell-type Transformer. 

or paper is sufficient to afford adequate insulation be- 
tween stampings. Ordinary iron of good magnetic 
quality may be used for transformers on the lower fre- 
quencies, but it is customary to use special alloyed 
iron for 6o-cycle transformers. This material has a 
high electrical resistance and, therefore, a small eddy- 
current loss. The loss through hysteresis is also small, 
but the permeability of alloyed iron is lower than that 
of ordinary iron and this tends to increase the magnetiz- 


ing current. The cost of alloyed iron is appreciably 
higher than that of ordinary transformer iron. 

The choice of type whether " core " or " shell " 
will not greatly affect the efficiency or cost of the trans- 
former. As a general rule, the core type of construc- 
tion has advantages in the case of high-voltage trans- 
formers of small output, while the shell type is best 
'adapted for low- voltage transformers of large output. 

Fig. 10 illustrates a good practical design of shell- type 
transformer in which a saving of material is effected 
by arranging the magnetic circuit to surround all four 
sides of a square coil. The dimensions of the iron cir- 
cuit, as indicated on the sketch, show a cross-section 
of the magnetic circuit outside the coils exactly double 
the cross-section inside the coils. This will be found to 
lead to slightly higher efficiency, for the same cost of 
material, than if the section were the same inside and 
outside the coil. It is generally advantageous to use 
higher flux densities in the iron upon which the coils 
are wound than in the remainder of the magnetic cir- 
cuit, because the increased iron loss is compensated for 
by the reduced copper loss due to the shorter average 
length per turn of the windings. 

Fig. ii illustrates a similar design of shell- type trans- 
former in which the magnetic circuit is still further 
divided, and the windings are in the form of cylindrical 
coils. The relative positions of primary and secondary 
coils need not be as shown in Figs. 10 and n, as they 
can be of the " pancake " shape of no great thickness, 
with primary and secondary coils alternating. A proper 
arrangement of the coils is a matter of great importance 


when it is desired to have as small a voltage drop as 
possible under load; but this point will be taken up 

FIG. 10. Shell-type Transformer with Distributed Magnetic Circuit. 
(Square core and coil.) 

again when dealing with magnetic leakage and regula- 
Fig. 12 illustrates a common arrangement of the 



stampings and windings in a three-phase core-type 
transformer. Each of the three cores carries both pri- 
mary and secondary coils of one phase. The portions 

FIG. ii. Shell-type Transformer with Distributed Magnetic Circuit. 
(Berry transformer with circular coil.) 

of the magnetic circuit outside the coils must be of 
sufficient section to carry the same amount of flux as 
the wound cores. This will be understood if a vector 
diagram is drawn showing the flux relations in the 


various parts of the magnetic circuit. This use of cer- 
tain parts of the magnetic circuit to carry the flux com- 
mon to all the cores leads to a saving in material on 
what would be necessary for three single-phase trans- 
formers of the same total k.v.a. output; but, as men- 

FIG. 12. Three-phase Core-type Transformer. 

tioned in Article 5, it does not follow that a three-phase 
transformer is always to be preferred to three separate 
single-phase transformers. 

Figs. 13 and 14 show sections through three-phase 
transformers of the shell type. The former is the more 



common design, and it has the advantage that rect- 
angular shaped stampings can be used throughout. 
The vector diagram in Fig. 13 shows how the flux 3> e 
in the portion of the magnetic circuit between two sets 
of coils has just half the value of the flux $ in the cen- 
tral core. 


FIG. 13. Section through Three-phase Shell Transformer, 
phase consists of one H.T. and two L.T. coils.) 


9. Mechanical Stresses in Transformers. The 

mechanical features of transformer design are not of 
sufficient importance to warrant more than a brief 
discussion. In the smaller transformers it is merely 
necessary to see that the clamps or frames securing 
the stampings and coils in position are sufficiently sep- 
arated from the H.T. windings, and that 'bolts in which 


e.m.f.'s are likely to be generated by the main or stray 
magnetic fluxes are suitably insulated to prevent the 
establishment of electric currents with consequent PR 
losses. The tendency in all modern designs is to avoid 
cast iron, and use standard sections of structural steel 
in the assembly of the complete transformer. In this 

FIG. 1 4. Special Design of Three-phase Shell- type Transformer. 

manner the cost of special patterns is avoided and a 
saving in weight is usually effected. The use of stand- 
ard steel sections also gives more flexibility in design, 
as slight modifications can be made in dimensions with 
very little extra cost. 

In large transformers, the magnetic forces exerted 
under conditions of heavy overloads or short-circuits 


may be sufficient to displace or bend the coils unless 
these are suitably braced and secured in position; and 
since the calculation of the stresses that have to be 
resisted belong properly to the subject of electrical 
design, it will be necessary to determine how these 
stresses can be approximately predetermined. 

The absolute unit of current may be defined as the 
current in a wire which causes one centimeter length 
of the wire, placed at right angles to a magnetic field, 
to be pushed sidewise with a force of one dyne when 
the density of the magnetic field is one gauss. 

Since the ampere is one- tenth of the absolute unit of 
current, we may write, 

' BIl 

where F = Force in dynes; 

B = Density of the magnetic field in gausses; 
/ = Current in the wire (amperes) ; 
/ = Length of the wire (centimeters) in a direction 
perpendicular to the magnetic field. 

It follows that the force tending to push a coil of wire 
of T turns bodily in a direction at right angles to a 
uniform magnetic field of B gausses (see Fig. 15) is 

17 BITl A 

p = _ dynes. 

If both current and magnetic field are assumed to 
vary periodically according to the sine law, passing 
through corresponding stages of their cycles at the 


same instant of time, we have the condition which is 
approximately reproduced in the practical transformer 
where the leakage flux passing through the windings is 
due to the currents in these windings. 

Coil of T wires, each 
-carrying I amperes 

FIG. 15. Force Acting on Coil-side in Uniform Magnetic Field. 

Since the instantaneous values of the current and 
flux density will be 7 max sin 0, and B max sin 8, respectively, 
the average mechanical force acting upon the coil may 
be written, 

Tl. i C* . TIL 


= /max^max I sill 





.2 fiflfi 




If the flux density is not uniform throughout the sec- 
tion of coil considered, the average value of 5 max should 
be taken. Let this average value of the maximum den- 
sity be denoted by the symbol B am . Then, since i Ib. = 
444,800 dynes, the final expression for the average 
force tending to displace the coil is, 

1 II m ax -Dam .. 

Force= w;~ lb 

In large transformers the amount of leakage flux 
passing through the coils may be considerable. It will 
be very nearly directly proportional to / max , and the 
mechanical forces on transformer coils are therefore 
approximately proportional to the square of the current. 
As the short-circuit current in a transformer which is 
not specially designed with high reactance might be 
thirty times the normal full-load current, the mechan- 
ical forces due to a short-circuit may be about 1000 
times as great as the forces existing under normal work- 
ing conditions. 

Except in a few special cases, the calculation of the 
leakage flux is not an easy matter, and the value of B &m 
in Eq. (4) cannot usually be predetermined exactly; 
but it can be estimated with sufficient accuracy for the 
purpose of the designer, who requires merely to know 
approximately the magnitude of the mechanical forces 
which have to be resisted by proper bracing of the coils. 

The calculation of leakage flux will be considered 
when discussing voltage regulation; but in the case of 
"sandwiched " coils as, for instance, in the shell type of 


:ransformer shown in Fig. 16, the distribution of the 
leakage flux will be generally as indicated by the dia- 
gram plotted over the coils at the bottom of the sketch. 





FIG. 1 6. Forces in Transformer Coils Due to Leakage Flux. 

When the relative directions of the currents in the 
primary and secondary coils are taken into account, it 



will be seen that all the forces tending to push the coilr 
sidewise are balanced, except in the case of the two 
outside coils. In each individual coil the effect of the 
leakage flux is to crush the wires together; but the end 

FIG. 17. Core-type Transformer with " Sandwiched " Coils. 

coils will be pushed outward unless properly secured in 

Since there is no resultant force tending to move the 
windings bodily relatively to the iron stampings, a 
simple form of bracing consisting of insulated bars and 


tie rods, as shown in Fig. 16 will satisfy all requirements, 
and this bracing can be quite independent of the frame- 
work or clamps supporting the transformer as a whole. 
In the case of core-type transformers, with rect- 
angular coils arranged axially one within the other, the 
mechanical forces will tend to force the coils into a cir- 
cular shape. With cylindrical concentric coils, no spe- 
cial bracing is necessary provided the coils are symmet- 
rically placed axially; but if the projection of one coil 
beyond the other is not the same at both ends, there 
will be an unbalanced force tending to move one coil 
axially relatively to the other. If the core type of 
transformer is built up with flat strip " sandwiched " 
coils, the problem is generally similar to that of the shell 
type of construction. A method of securing the end coils 
in position with this arrangement of windings is illus- 
trated by Fig, 17, 



10. The Dielectric Circuit. Serious difficulties are not 
encountered in insulating machinery and apparatus 
for working pressures up to 10,000 or 12,000 volts, but 
for higher pressures (as in 150,000- volt transformers) 
designers must have a thorough understanding of the 
dielectric circuit,* if the insulation is to be correctly 
and economically proportioned. The information here 
assembled should make the fundamental principles of 
insulation readily understood and should enable an 
engineer to determine in any specific design of trans- 
former the thicknesses of insulation required in any 
particular position, as between layers of windings, 
between high-tension and low- tension coils, and be- 
tween high-tension coils and grounded metal. The data 
and principles outlined should also facilitate the deter- 
mination of dimensions and spacings of high-tension 
terminals and bushings of which the detailed design is 
usually left to specialists in the manufacture of high- 
tension insulators. In presenting this information two 
questions are considered: (i) What is the dielectric 

* " Insulation and Design of Electrical Windings," by A. P. M 
Fleming and R. Johnson Longmans, Green & Co. 

" Dielectric Phenomena in High- voltage Engineering," by F. W. 
Peek, Jr. McGraw-Hill Book Company, Inc. 



strength of the insulating materials used in transformer 
design? and (2) how can the electric stress or voltage 
gradient be predetermined at all points where it is liable 
to be excessive? 

Apart from a few simple problems of insulation 
capable of a mathematical solution, the chief difficulty 
encountered in practice usually lies in determining the 
distribution of the dielectric flux, the concentration of 
which at any particular point may so increase the fkix 
density and the corresponding electric stress that dis- 
ruption of the dielectric may occur. The conception of 
lines of dielectric flux, and the treatment of the dielec- 
tric circuit in the manner now familiar to all engineers 
in connection with the magnetic circuit has made it pos- 
sible to treat insulation problems * in a way that is 
equally simple and logical. 

The analogy between the dielectric and magnetic 
circuits may be illustrated by Fig. 18, where a metal 
sphere is supposed to be placed some distance away from 
a flat metal plate, the intervening space being occupied 
by air, oil, or any insulating substance of constant 
specific capacity. This arrangement constitutes a con- 
denser of which the capacity is (say) C farads. If a 
difference of potential of E volts is established between 

* The dielectric circuit is well treated from this point of view in 
the following (among other) books: 

"The Electric Circuit," by V. Karapetoff McGraw-Hill Book 
Company, Inc. 

" Electrical Engineering," by C. V. Christie McGraw-Hill Book 
Company, Inc. 

" Advanced Electricity and Magnetism," by W. S. Franklin and 
B. MacNutt Macmillan Company. 


the sphere and the plate, the total dielectric flux, 
^ will have to satisfy the equation 

* = EC, (5) 

where ^ is expressed in coulombs, E in volts, and C in 

The quantity M> coulombs of electricity should not be 
considered as a charge which has been carried from the 
sphere to the plate on the surface of which it remains, 
because the whole of the space occupied by the dielectric 
is actually in a state of strain, like a deflected spring, 
ready to give back the energy stored in it when the 
potential difference causing the deflection or displace- 
ment is removed. Instead, the dielectric should be 
considered as an electrically elastic material which will 
not break down or be ruptured until the " elastic limit '' 
has been reached. The quantity SF, which is called the 
dielectric flux, may be thought of as being made up of 
a definite number of unit tubes of induction, the direc- 
tion of which in the various portions of the dielectric 
field is represented by the full lines in Fig. 18. The 
name of the unit tube of dielectric flux is the coulomb. 

If the sphere were the north pole and the plate the 
south pole of a magnetic circuit, the distribution of 
flux lines would be similar. The total flux would then 
be denoted by the symbol $, and the unit tube of induc- 
tion would be called the maxwell. In place of formula 
(5) the following well-known equation could then be 
written : 

< = Mmf X permeance (6) 


This expression is analogous, to the fundamental 
equation for a dielectric circuit, the electrostatic capacity 
C being, in fact, a measure of the permeance of the di- 
electric circuit, while , sometimes called the elastance, 

may be compared with reluctance in the magnetic 

The dotted lines in Fig. 18 are sections through equi- 
potential surfaces. The potential difference between 


FIG. 1 8. Distribution of Dielectric Flux between Sphere and Flat Plate. 

any two neighboring surfaces, as drawn, is one-quarter 
of the total. At ,all points the lines of force, or unit 
tubes of induction, are perpendicular to the equipoten- 
tial surfaces. Furthermore, the flux density, or cou- 
lombs per square centimeter, through any small portion 
A of an equipotential surface over which the distribu- 
tion may be considered practically uniform is 




The capacity, or permittance* of a small element of 
the dielectric circuit of length / and cross-section A 

is proportional to --, or with the proper constants 


Electrostatic capacity = C = ( , lo9 -W farads (8) 

\47r(3Xio 10 ) 2 / / 

wherein the numerical multiplier results from the choice 
of units. The factor k is the specific inductive capacity, 
or dielectric constant, of the material (k = i in air), 
while the unit for / and A is the centimeter. This ex- 
pression for capacity may conveniently be rewritten as 

' 8.84 kA . t 
mf = ^ microfarads. ... (9) 

Values of k are given in the accompanying table to- 
gether with the dielectric strengths of the materials. 
These figures are only approximate, those referring to 
dielectric strength merely serving as a rough indication 
of what the material of aveiage quality may be expected 
to withstand. The figures indicate the approximate 
virtual or r.m.s. value of the sinusoidal alternating 
voltage which, if applied between two large flat elec- 
trodes, would lead to the breakdown of a i-cm. slab 
of insulating material placed between the electrodes. 

What is generally understood by the disruptive gra- 
dient, or stress in kilovolts per centimeter, would be 

* The reciprocal of elastance. 


about \/~2 times the value given in the last column of 
the table. Thus, if a battery or continuous-current 
generator were used in the test, the pressure necessary to 
break down a 0.75-011. film of air between two large flat 
parallel plates would be loooXV^X 22X0. 75 = 23,400 




Kv. per Cm. 




Transformer oil 

2 3 

Paper (dry) 



Paper (oil impregnated) 


Pressboard (dry or varnished). . . 



Pressboard (oil impregnated) 


I ?O 

Treated wood (across grain) 

3 to 6 


Varnished cambric 





6 to 8 



^ to 7 



4 to 10 




Returning to Formula (7), let electric flux or quantity 
of electricity, ^, be expressed in terms of capacity and 
e.m.f., with a view to determining the relation between 
flux density and electric stress. The Formula (8) may 
be written 

C=Kkj farads, 


where K stands for the numerical constant. Sub- 
stituting in Formula (5), 


Since is the potential gradient, or voltage drop 

per centimeter, which is sometimes referred to as the 
electrostatic force or electrifying force, and denoted by 
the symbol G, we may write, 

D = KkxG. . . , . . (10) 
The analogous expression for the magnetic circuit is, 

In the case of a dielectric circuit, electric flux density 

= e.m.f. per centimeter X " conductivity " of the material 

to dielectric flux, while in the magnetic circuit, magnetic 

flux density = m.m.f. per centimeter X " conductivity " 

of the material to magnetic flux. 

Since the electric stress or voltage gradient G is 
directly proportional (in a given material) to the flux 
density D, it follows that when the concentration of 
the flux tubes is such as to produce a certain maximum 
density at any point, breakdown of the insulation will 
occur at this point. Whether or not the rupture will 
extend entirely through the insulation will depend upon 


the value of the flux density (consequently the potential 
gradient) immediately beyond the limits of the local 

Given two electrical conductors of irregular shape, 
separated by insulating materials, the problem of cal- 
culating the capacity of the condenser so formed is 
very similar to that of calculating the permeance of 
the magnetic paths between two pieces of iron of very 
high permeability separated by materials of low per- 
meability. There is no simple mathematical solution 
to such a problem, and the best that can be done is to 
fall back on the well-established law of maximum per- 
meance, or " least resistance." According to this law 
the lines of force and equipotential surfaces will be so 
shaped and distributed that the permittance, or capacity, 
of the flux paths will be a maximum. With a little 
experience, ample time, and a great deal of patience, 
the probable field distribution can generally be 
mapped out, even in the case of irregularly shaped 
surfaces, with sufficient accuracy to emphasize 
the weak points of the design and to permit of 
the maximum voltage gradient being approximately 

Before illustrating the application of the above prin- 
ciples in the design of transformer insulation, it will 
be advisable to assemble and define the quantities which 
are of interest to the engineer in making practical 

* This method of plotting flux lines is explained, in connection with 
the magnetic field, at some length in the writer's book " Principles 
of Electrical Design." McGraw-Hill Book Co., Inc. 



E, e = e.m.f . or potential difference (volts) ; 

/ = length, measured along line of force (centimeters) ; 
yl=Area of equipotential surface perpendicular to 
lines of force (square centimeters) ; 


G = -r, = potential gradient (volts per centimeter); 

C = Capacity or permittance (farads) ; 

(farads = ,- = nux per unit e.m.f.); 

K constant = 8.84 X io~ 14 (farads per centimeter cube, 
being the specific capacity of air) ; 

k = dielectric constant, or relative specific capacity, 
or permittivity (k i for air) ; 

^ = dielectric flux, or electrostatic induction (^ = 
CE AD coulombs) ; 


Z) = flux density = -r = KkG (coulombs per square centi- 

11. Capacity of Plate Condenser. Imagine two par- 
allel metal plates, as in Fig. 19, connected to the oppo- 
site terminals of a direct-current generator or battery. 
The area of each plate is A square centimeters and the 
separation between plates is / centimeters, the dielectric 
or material between the two surfaces being air. The 
edges of the plates should be rounded off to avoid con- 
centration of flux lines. If the area A is large in com- 
parison with the distance /, a uniform distribution of the 
flux ^ may be assumed in the air gap, the density being 



By Formula (9) the capacity is C m / = 



farads, since the specific capacity of air (k) is i. As- 
suming numerical values, let A = 1000 sq. cm., and 

= o.5 cm. Then, C = - ~ = i.yyXio- 10 farads. 


If - = 10,000 volts, the potential gradient will be 
G = = 20,000 volts per centimeter. There will be 


Distanced = 0.5 cm, 

FIG. 19. Flat Electrodes Separated by Air. 

no disruptive discharge, however, because a gradient 
of 31,000 volts per centimeter is necessary to cause 
break-down in air. 

By Formula (5) the total dielectric flux is ^ = 10,000 
Xi.77Xio- 10 = i,77Xio- 6 coulombs. 

Charging Current with Alternating Voltage. The effect 
of an alternating e.m.f., the crest value of which is 10,000 
volts, would be to displace the above quantity of elec- 
tricity 4/ times per second, / being the frequency. 


The quantity of electricity can be expressed in terms 
of current and time, thus, quantity = current X time, or 
coulombs = average value of current (in amperes) during 
quarter peri o d X time (in seconds) of one quarter period. 

/ 2 V~2\ I 

Therefore, ^ = /X|- -) :., where / stands for the 

\ TT 747 

virtual or r.m.s. value of the charging current on the 

sine wave assumption. Transposing terms, I = ~. 

2V 2 

If E is now understood to stand for the virtual value 
of the alternating potential difference, ty = CExv / 2, 
whence I = 2irfCE, which is the well-known formula 
for calculating capacity current on the assumption of 
sinusoidal wave shapes. 

12. Capacities in Series. When condensers are con- 
nected in parallel on the same source of voltage, the 
total dielectric flux is evidently determined by summing 
up the fluxes as calculated or measured for the individual 
condensers. In other words, the total capacity is the 
sum of the individual capacities. With condensers in 
series, however, the total flux, or displacement, will be 
the same for all the capacities in series, therefore, the 
calculations may be simplified just as for electric or 
magnetic circuits by adding the reciprocals of the con- 
ductance or permeance. The conception of elastance, 
corresponding to resistance in the electric circuit and 
reluctance in the magnetic circuit, is thus seen to 
have certain advantages. In the dielectric circuit 

Elastance = 

permittance (or capacity) C' 


For a concrete example, assume that a 0.3-011. plate 
of glass is inserted between the electrodes of the con- 
denser shown in Fig. 19. The modified arrangement is 
illustrated by Fig. 20. On first thought it might appear 
that this arrangement would improve the insulation, 
but care must always be taken when putting layers of 
insulating materials of different specific inductive capac- 
ity in series, as this example will illustrate. In addition 
to the elastance of a 0.3 -cm. layer of glass there is the 

0.3 cm. thick. 

FIG. 20. Electrodes Separated by Air and Glass. 

elastance of two layers of air of which the total thickness 
is 0.2 cm. Assuming that the value of the dielectric 
constant k for the particular quality of glass used is 
7 and that G g and G a are the potential gradients in 
the glass and air respectively, then, by formula (10) 
KG a yKGo, whence G a jGg. 

Taking the total potential difference between elec- 
trodes as 10,000 volts, the same as used in considering 
Fig. 19, E = 10,000 = o.2G a +0.360, whence G g = 5880 volts 



per centimeter, and G a = 41,100 volts per centimeter. 
Such a high gradient as 41,100 would break down the 
layers of air and would manifest itself by a bluish elec- 
trical discharge between the metal plates and the glass. 
On the other hand, the gradient of 5880 volts per cen- 
timeter would be far below the stress necessary to 
rupture the glass. Nevertheless a discharge across air 
spaces should always be avoided in practical designs 
because of its injurious effect on the metal surfaces and 
also on certain types of insulating material. It should 
be observed that the introduction of the glass plate 
has appreciably increased the capacity of the con- 
denser. For example, with the same voltage (E = 10,000) 
as before, the total flux is now^ = A D = 1000 (8.84X io~ 14 
X4i,ioo) =3.63Xio~ 6 coulombs. This value is about 
double the value calculated with only air between the 
condenser plates. 

As a practical application of the principles governing 
the behavior of condensers in series, consider the insu- 
lation between the coils and core of an air-cooled trans- 
former, i.e., of which the coils are not immersed in oil. 
In addition assume the insulation to consist of layers of 
different materials made up as follows: 






Constant, k. 

Cotton braiding and varnished cambric 
Micanite . . 



O 3I7 




o 158 

Air spaces (estimated) 


o 061 



Then, suppose it is desired to determine how high 
an alternating voltage can be applied between the coils 
and the core before the maximum stress in the air 
spaces exceeds 31,000 volts per centimeter, the gradient 
which will cause disruption and static discharge, with 
the consequent danger to the insulation due to local 
heating and chemical action. Assuming the coil to 
constitute one flat plate of a condenser of which the 
other plate is the iron frame or core, the effect is that 
of a number of plate condensers in series the total elas- 
tance being 

- = + ++--. 
C C\ 2, Ca C 

By Formula (8), the individual capacities for the 

same surface area are proportional to j, and 


C ki J?2 k$ k 



C '' * = ~D~~KG ai r~Gair J 

the permissible maximum value of E is 

= 6260 volts (maximum). 

The r.m.s. value of the corresponding sinusoidal alter- 

natmg voltage is =-=4430, which is the limiting 



potential difference between windings and grounded 
metal work if the formation of corona is to be avoided. 
A transformer having insulation made up as previously 
described would be suitable for a 66oo-volt three-phase 
circuit with grounded neutral; but for higher voltages 
the insulation should be modified, or oil immersion 
should be employed to fill all air spaces. If the oil- 
cooled construction is employed, the previously con- 
sidered insulations (slightly modified in view of pos- 
sible action of the oil upon the varnish) would probably 
be suitable for working voltages up to 15,000. 

13. Surface Leakage. A large factor of safety must 
be allowed when determining the distance between 
electrodes measured over the surface of an insulator. 
Whether or not spark-over will occur depends not only 
upon the condition of the surface (clean or dirty, dry 
or damp), but also upon the shape and position of the 
terminals or conductors. It is therefore almost impos- 
sible to determine, other than by actual test, what will 
happen in the case of any departure from standard 
practice. Surface leakage occurs under oil as well as 
in air, but generally speaking, the creepage distance 
under oil need be only about one-quarter of what is 
necessary in air. 

An important point to consider in connection with 
surface leakage is illustrated by Figs. 21 and 22. In 
Fig. 21, a thin disk of porcelain (or other solid insulator) 
separates the two electrodes, while in Fig. 22, the same 
material is in the form of a thick block providing a 
leakage path (/) of exactly the same length as in Fig. 21. 
The voltage required to cause spark-over will be con- 


siderably greater for the block of Fig. 22 than for the 
disk of Fig. 21. This condition exists because the flux 
concentration due to the nearness of the terminals in 
Fig. 21 begins breaking down the layers of air around 
the edges of the electrodes at a much lower total poten- 
tial difference than will be necessary in the case of the 
thicker block of Fig. 22. The effect of the incipient 
breakdown is, virtually, to make a conductor of the air 

FIG. 21. 

FIG. 22. 

FIG. 21. Surface Leakage over Thin Plate. 

FIG. 22. Surface Leakage Over Thick Insulating Block. 

around the edges of the metal electrodes, and a very 
slight increase in the pressure will often suffice to break 
down further layers of air and so result in a discharge 
over the edges of the insulating disk. The phenomenon- 
of so-called surface leakage may thus be considered 
as largely one of flux concentration or potential gra- 
dient. Sometimes it will be easier to eliminate trouble 
due to surface leakage by altering the design of ter- 


minals and increasing the thickness of the insulation 
than by adding to the length of the creepage paths. 

14. Practical Rules Applicable to the Insulation of 
High-voltage Windings. For working pressures up to 
16,000 volts, solid insulation, including cotton tape, 
micanite, pressboard, horn paper, or any insulating 
material of good quality used to separate the windings 
from the core or framework, should have a total thick- 
ness of approximately the following values: 

Voltage. Thickness of Insulation (Mils) 

IIO 4 

400 45 

1,000 65 

2,200 90 

6,600 180 

12,000 270 

16,000 350 

In large high- volt age power transformers, cooled by 
air blast, the air spaces are relied upon for insulation. 
The clearances between coils and core or case are neces- 
sarily much larger than in oil-cooled transformers, and 
calculations similar to the example previously worked 
out should be made to determine whether or not the 
insulation is sufficient and suitably proportioned to 
prevent brush discharge. The calculations are made 
on the basis of several plate condensers in series; thus 
the flux density and dielectric stress in the various 
layers of insulation can be approximately predetermined. 
The difficulty of avoiding static discharges will generally 


stand in the way of designing economical air-cooled 
transformers for pressures much in excess of 30,000 
volts. A rough rule ior air clearance is to allow a 

distance equal to inches, where kv stands for 


the virtual value of the alternating potential differ- 
ence in kilovolts between the two surfaces considered. 

With oil-immersed transformers, the oil channels 
should be at least 0.25 in. wide in order that there may 
be free circulation of the oil. In high-voltage trans- 
formers having a considerable thickness of insulation 
between coils and core, it is advantageous to divide the 
oil spaces by partitions of pressboard or similar mate- 
rial. Assuming the total thickness of oil to be - no 
greater than that of the solid insulation, a safe rule is 
to allow i mil for every 25 volts. For instance, a total 
thickness of insulation of i in. made up of 0.5 in. of 
solid insulation and two 0.25 in. oil ducts would be suit- 
able for a working pressure not exceeding 25X1000 = 
25,000 volts. Further particulars relating to oil insula- 
tion will be given later. 

It is customary to limit the volts per coil to 5000, and 
the volts between layers of winding to 400. Special 
attention must be paid to the insulation under the 
finishing ends of the layers by providing extra insula- 
tion ranging from thin paper to Empire cloth or even 
thin fullerboard, the material depending upon the voltage 
and also upon the amount of mechanical protection 
required to prevent cutting through the insulation where 
the wirer cross. Sometimes the insulation is bent 
around the end wires of a layer to prevent breakdown 


over the ends of the coil. Where space permits, however, 
the layers of insulation may be carried beyond the ends of 
the winding so as to avoid surface leakage. This arrange- 
ment is more easily carried out in core-type transformers 
than in shell-type units. A practical rule for deter- 
mining the surface distance (in inches) required to pre- 
vent leakage (given by Messrs. Fleming and Johnson 
in the book previously referred to) is "to allow 0.5 in. 
+o.5X kilovolts, when the surfaces are in air. For sur- 
faces under oil, the allowance may be 0.5+0.1 Xkilovolts. 
In any case it is important to see that the creepage sur- 
faces are protected as far as possible from deposits of 
dirt. When the coils of shell-type transformer are 
"sandwiched," it is customary to use half the normal 
number of turns in the low-tension coils at each end of 
the stack. This has the advantage of keeping the high- 
tension coils well away from the iron stampings and 
clamping plates or frame. 

Extra Insulation on End Turns. Concentration of 
potential between turns at the ends of the high-tension 
winding is liable to occur with any sudden change of 
voltage across the -transformer terminals, such as when 
the supply is switched on, or when lightning causes 
potential disturbances on the transmission lines. It is, 
therefore, customary to pay special attention to the insu- 
lation of the end turns of the high-tension winding. 
Transformers for use on high-voltage circuits usually 
have about 75 ft. at each end of the high-tension winding 
insulated to withstand three to four times the voltage 
between turns that would puncture the insulation in the 
body of the winding. 


It is very difficult to predetermine the extra pressure 
to which the end turns of a power transformer con- 
nected to an overhead transmission line may at times 
be subjected, but it is safe to say that the instantaneous 
potential difference between turns may occasionally be 
of the order of forty to fifty times the normal working 
pressure. In such cases the usual strengthening of the 
insulation on the end turns would not afford adequate 
protection, and for this reason a separate specially 
designed reactance coil connected to each end of the high- 
tension winding would seem to be the best means of 
guarding against the effects of surges or sudden changes 
of pressure occurring in the electric circuit outside the 
transformer. The theory of abnormal pressure rises in 
the end sections of transformer windings will not be 
discussed here. 

15. Winding Space Factor. Knowing the thickness 
of the cotton covering on the wires, the insulation 
between layers of winding, between coil and coil 
and between coil and iron stampings, it becomes 
an easy matter to determine approximately the 
total cross-section of the winding-space to accom- 
modate a given cross-section of copper. The ratio 

cross-section of copper 

, which is known as the 
cross-section of winding space 

space factor, will naturally decrease with the higher 
voltages and smaller sizes of wire. This factor may be 
as high as 0.46 in large transformers for pressures not 
exceeding 2200 volts; in 33,000- volt transformers for 
outputs of 200 k.v.a. and upward it will have a value 
ranging between 0.35 and 0.2, while in oil-immersed 


power transformers for use on ioo,ooo-volt circuits the 
factor may be as low as 0.06. 

16. Oil Insulation. There is a considerable amount of 
published matter relating to the properties of insulating 
oils, and also to the various methods of testing, puri- 
fying, and drying oils for use in transformers. A con- 
cise statement of the points interesting to those installing 
or having charge of transformers will be found in W. T. 
Taylor's book on transformers.* What follows here is 
intended merely as a guide to the designer in providing 
the necessary clearances to avoid spark-over, including a 
reasonable factor of safety. 

Mineral oil is generally employed for insulating pur- 
poses, its main function in transformers being to trans- 
fer the heat by convection from the hot surfaces to the 
outside walls of the containing case, or to the cooling 
coils when these are provided. The presence of an 
extremely small percentage of water reduces the insu- 
lating properties of oil considerably. It is therefore 
important to test transformer oil before using it, and if 
necessary extract the moisture by filtering through dry 
blotting paper, or by any other approved method. Dry 
oil will withstand pressures up to 50,000 volts (alter- 
nating) between brass disks 0.5 in. in diameter with a 
separation of 0.2 in. For use in high- voltage trans- 
formers, the oil should be required to withstand a test 

* " Transformer Practice," by W. T. Taylor McGraw-Hill Book 
Company, Inc. For further information refer H. W. Tobey on the 
"Dielectric Strength of Oil " Trans. A.I.E.E.; Vol. XXIX, page 
1189 (1910). Also " Insulating Oils," Journ. Inst. E.E.. Vol. 54, page 
497 (1916). 


of 45,000 volts under the above conditions. The good 
insulating qualities of oil suggest that only small clear- 
ances would be required in transformers, even for high 
voltages; but the form of the surfaces separated by the 
layer of oil will have a considerable effect upon the con- 
centration of flux density, and therefore upon the volt- 
age gradient. As an example, if 100,000 volts breaks 
down a i -in. layer of a certain oil between two parallel 
disks 4 in. in diameter, the same pressure will spark 
across a distance of about 3.5 in. between a disk and a 
needle point. 

Partitions of solid insulation such as pressboard or 
fullerboard are always advisable in the spaces occu- 
pied by the oil, since they will prevent the lining up of 
partly conducting impurities along the lines of force 
and reduce the total clearance which would otherwise 
be necessary. 

In a transformer oil of average quality, the sparking 
distance between a needle point and a flat plate is approx- 
imately (o.25+o.o4Xkv.) inches. Since there may 
be sharp corners or irregularities corresponding to a 
needle point, which will produce concentration of 
dielectric flux, it therefore seems advisable to introduce 
a factor of safety for oil spaces between high tension 
and grounded metal fgr instance, between the ends of 
high-tension coils and the containing case by basing 
the oil space dimension on the formula, 

Thickness of oil (inches) =0.25+0.08 Xkv., . (n) 
where kv. stands for the working pressure in kilovolts. 


With two or three partitions of solid insulating mate- 
rial dividing the oil space into sections, the total 
ness need not exceed 

0.25+0.05 Xkv (12) 

If the total thickness of solid insulation is about equal 
to that of the oil ducts (not an unusual arrangement 
between coils and core), the rule previously given for 
solid insulation may be slightly modified to include a 
minimum thickness of 0.25 in., and put in the form, 

Total thickness of oil ducts plus ) 

solid insulation of app'roxi- I =0.2 5 +0.03 Xkv. (13) 

mately equal thickness (inches) J 

A suitable allowance for surface leakage under oil, in 
inches, as already given, is 

0.5 +0.1 Xkv. . o, . . , (14) 

17. Terminals and Bushings. The exact pressure 
which will cause the breakdown of a transformer ter- 
minal bushing generally has to be determined by test, 
because the shape and proportions of the metal parts 
are rarely such that the concentration of flux density 
at corners or edges can be accurately predetermined.* 

* The reader who desires to go deeply into the study of high-pressure 
terminal design should refer to the paper by Mr. Chester W. Price 
entitled " An Experimental Method of Obtaining the Solution of Elec- 
trostatic Problems, with Notes on High-voltage Bushing Design." Trans. 
A.I.E.E., Vol. 36, page 905 (Nov., 1917). 


However, there are certain important points to bear 
in mind when designing the insulation of transformer 
terminals, and these will now be referred to briefly. 

The high-tension leads of a transformer may break 
down (i) by puncture of the insulation, or (2) by 
spark-over from terminal to case. If the transformer 
lead could be considered as an insulated cable with a 
suitable dielectric separating it from an outer concentric 

FIG. 23. Section through Insulated Conductor. 

metal tube of considerable length, the calculation of 
the puncture voltage (i) would be a simple matter. 
For instance, let r in Fig. 23 be the radius of the inner 
(cylindrical) conductor, and R the internal radius of 
the enclosing tube, the space between being filled with 
a dielectric of which the specific inductive capacity 
(k) is constant throughout the insulating material. 
The equipotential surfaces will be cylinders, and the 


flux density 'over the surface of any cylinder of radius 

x and of length i cm., will be D = . 


By Formula (10) the potential gradient is, 
D ^f 

In order to express this relation in terms of the total 
voltage E, it is necessary to substitute for the symbol 
^ its equivalent ExC, and calculate the capacity 
C of the condenser formed by the rod and the con- 
centric tube. Considering a number of concentric shells 
in series, the elastance may be written as follows : 

i r 
C = J 

dx i R 

C = 

Substituting in (15), we have, 


G = jp volts per centimeter, . . (17) 

x log, 

the maximum value of which is at the surface of the 
inner conductor, where 


This formula is of some value in determining the thick- 
ness of insulation necessary to avoid overstressing the 


dielectric; but it is not strictly applicable to trans- 
former bushings in which the outer metal surface (the 
bushing in the lid of the containing tank) is short in 
comparison with the diameter of the opening. The 
advantage of having a fairly large value for r is indicated 
by Formula (18), and a good arrangement is to use a 
hollow tube for the high-tension terminal, with the lead 
from the windings passing up through it to a clamping 
terminal at the top. 

Solid porcelain bushings with either smooth or cor- 
rugated surfaces may be used for any pressure up to 
40,000 volts, but for higher pressures the oil-filled type 
or the " condenser " type of terminal is preferable. In 
designing plain porcelain bushings it is important to 
see that the potential gradient in the air space between 
the metal rod and the insulator is not liable to cause 
brush discharge, as this would lead to chemical action, 
and a green deposit of copper nitrate upon the rod. The 
calculations would be made as explained for the parallel- 
plate condensers in which a sheet of glass was inserted 
(see " Capacities in Series "), except that the elastances 
of the condensers are now expressed by Formula (16). 

18. Oil-filled Bushings. The chief advantages of a 
hollow insulating shell filled with oil or insulating com- 
pound that can be poured in the liquid state, are the 
absence of air spaces where corona may occur, and the 
possibility of obtaining a more uniform and reliable 
insulation than with solid insulators such as porcelain, 
when the thickness is considerable. The metal ring 
by which such an insulator (see Fig. 24) is secured to the 
transformer cover usually takes the form of a cylinder 


of sufficient length to terminate below the surface of the 
oil. The advantage of this arrangement is that the 
dielectric flux over the surface of the lower part of the 
insulator is through oil only, and not as would otherwise 
be the case, through oil and air. With the two mate- 
rials of different dielectric constants, the stress at the 
surface of the oil may exceed the dielectric strength of 
air, in which case there would be corona or brush dis- 
charge which might practically short-circuit the air 
path and increase the stress over that portion of the 
surface which is under the oil. 

The bushing illustrated in Fig. 24 has been designed 
for a working pressure of 88,000 volts between high- 
tension terminal and case, the method of computation 
being, briefly, as follows: Applying the rule for sur- 
face leakage distances previously given, this dimension 
is found to be 0.5 +- = 44. 5 in. The insulator need 
not, however, measure 44.5 in. in height above the 
cover of the transformer case, because corrugations can 
be used to obtain the required length. A safe rule to 
follow in deciding upon a minimum height, i.e., the 
direct distance in air between the terminal and the 
grounded metal, is to make this dimension at least as 
great as the distance between needle points that would 
just withstand the test voltage without sparking over. 
The test pressure is usually twice the working pressure 
plus 1000 volts, or 177 kv. (r.m.s. value) in this par- 
ticular case. This value corresponds to a distance of 
about 48 cm., or (say) 19 in. In order that there may 
be an ample margin of safety, it will be advisable to 
make the total height of the insulator not less than 22 



Iron Sleeve carried 
below surface of oil 

Metal tube of 

234 outside diaru 

ji Insulating tube around 
ifmetal cap and transformer 
I' lead 

H.T. Lead 

FIG. 24. Three-part Composition-filled Porcelain Transformer Bushing, 
Suitable for a Working Pressure of 88,000 Volts to Ground. 


in., apart from the number or depth of the corrugations. 
The actual height in Fig. 24 is 31 in. because the cor- 
rugations on the outside of the porcelain shell are neither 
very numerous nor very deep. In this connection it 
may be stated that a short insulator with deep corruga- 
tions designed to provide ample surface distance is not 
usually so effective as a tall insulator with either a 
smooth surface or shallow corrugations. The reason 
is that much of the dielectric flux from the high-tension 
terminal to the external sleeve or supporting framework 
passes through the flanges, the specific inductive capacity 
of which is two to three times that of the air between 
them. The result is an increased stress in the air spaces, 
which is equivalent to a reduction in the effective height 
of the insulator. 

In the design under consideration it is assumed that 
the hollow (porcelain) shell is filled with an insulating 
compound which is solid at normal temperatures, and 
that the joints therefore need not be so carefully made 
as when oil is used. The insulator consists of three 
parts only, which are jointed as indicated on the sketch. 
Oil-filled bushings for indoor use generally have a large 
number of parts, usually in the form of flanged rings 
with molded tongue-and-groove joints filled with a 
suitable cement. There is always the danger, however, 
that a vessel so constructed may not be quite oil- tight, 
therefore the solid compound has an advantage over the 
oil in this respect. 

The creepage distance over the surface of the insulator 
in oil may be very much less than in air. Applying the 
rule previously given, the minimum distance in this 


case would be 0.5 + (0.1 X88) =9.3 in. In the design 
illustrated by Fig. 24, however, this dimension has been 
increased about 50 per cent with a view to keeping 
the high-tension connections well away from the sur- 
face of the oil and grounded metal. To prevent the 
accumulation of conducting particles in the oil along the 
lines of stress, and afford increased protection with only 
a small addition in cost, it is advisable to slip one or 
more insulating tubes over the lower part of the ter- 
minal, as indicated by the dotted lines in the sketch. 
Corrugations on the surface of the insulator in the oil 
are usually unnecessary, and sometimes objectionable 
because they collect dirt which may reduce the effective 
creepage distance. 

Having decided upon the height and surface distances 
to avoid all danger of spark-over, the problem which 
remains to be dealt with is the provision of a proper thick- 
ness of insulation to prevent puncture. In order to 
avoid complication of the problem by considering the 
different dielectric constants (k) of the compound used 
for filling and of the external shell (assumed in this 
case to be porcelain), it may be assumed either that 
there is no difference in the dielectric constants of the 
two materials, or that the thickness of the inclosing 
shell of porcelain is negligibly small in relation to the 
total external diameter of the insulator. Either assump- 
tion, neglecting the error due to the limited length of the 
external metal sleeve,* permits the use of Formula (18), 

* The maximum stress in the dielectric might be 5 to 10 per cent 
greater than calculated by using formulas relating to very long cylin- 
ders. The corners at the ends of the outer cylinder should be rounded 
off to avoid concentration of dielectric flux at these places. 


giving the relation between the maximum potential 
gradient and the dimensions of the bushing, without 

Suppose that the disruptive gradient of the insulating 
compound is 90 kv. per centimeter (maximum value) or 
63.5 kv. per centimeter (r.m.s. value) of the alternating 
voltage. With a test pressure of 177 kv. and a margin of 
safety of 25 per cent, the value of E in Formula (18) 
will therefore be =177X1. 25X^2 =313 kv. 

Since the disadvantage of a very small value of r is 
evident from an inspection of the formula, the outside 
diameter of the inner tube is made 2.25 in. Then, since 

G= E 

r log e - 

logio- = - -^- = 1.216, 

r 2.54X1.125X90X2.303 

whence ^ = 3.79, or (say) 3.75 in. An external diam- 
eter of 7.5 in. at the center of the insulator will there- 
fore be sufficient to prevent the stress at any point 
exceeding the rupturing value even under the test pres- 

19. Condenser Type of Bushing. If the total thick- 
ness of the insulation between the high-tension rod and 
the (grounded) supporting sleeve is divided into a num- 
ber of concentric layers by metallic cylinders, the con- 
centration of dielectric flux at certain points (leading to 
high values of the voltage gradient) is avoided. The 
bushing then consists of a number of plate condensers 
in series, with a definite potential difference between 


the plates. If the total radial depth of insulation is 
divided into a large number of concentric layers (of 
the same thickness), separated by cylinders of tinfoil 
(of the same area) , the several condensers would all have 
the same capacity. The dielectric flux density, and 
therefore the potential gradient, would then be the same 
in all the condensers, so that the outer layers of insula- 
tion would be stressed to the same extent as the inner 
layers, and the total radial depth of insulation would 
be less than when the stress distribution follows the 
logarithmic law (Formula 18) as in the case of the solid 
porcelain, or oil-filled, bushing. 

The section on the right-hand side of Fig. 25 is a 
diagrammatic representation of a condenser bushing 
shaped to comply with the assumed conditions of equal 
thicknesses of insulation and equal areas of the con- 
denser plates. With a sufficient number of concentric 
layers, the condition of equal potential difference be- 
tween plate and plate throughout the entire thickness 
would be approximated; but the creepage distance over 
the insulation between the edges of the metal cylinders 
would be much smaller for the outer layers than for 
layers nearer to the central rod or tube. It is equally, 
if not more, important to prevent excessive stress over 
the surface than in the body of the insulator, and a 
practical condenser type of terminal can be designed 
as a compromise between the two conflicting require- 
ments. By making the terminal conical in form, as 
indicated by the dotted lines on the right-hand side 
and the full lines on the left-hand side of the sketch 
(Fig. 25), neither of the ideal conditions will be exactly 



fulfilled, but practical terminals so constructed are easily 
manufactured, and give satisfaction on circuits up to 

Metal shield to control 
^^distribution of dielectric field. 

^H ^^^^p^^^^^ 

100,000 Volts I 





60,000 Volts 





40,000 Volts 




\^ metal 

20,000 ^ 

Zero potential ; 



\ // 


al tube or rod, 
ling H.T. lead, 


FIG. 25. Illustrating Principle of Condenser Type Bushing. 

150,000 volts. By varying the thickness of the indi- 
vidual insulating cylinders, it is an easy matter to 
design a condenser type terminal of which the con- 


densers in series all have the same capacity even while 
the outside surface is conical in shape as shown on the 
left-hand side of Fig. 25. This gives a uniform potential 
gradient along the surface, and results in a good practical 
form of condenser-type bushing. 

If the ends of the metal cylinders coincide with equi- 
potential surfaces having the same potential as that 
which they themselves attain by virtue of the respective 
capacities of the condensers in series, there will . be no 
corona or brush discharge at the edges of these cylin- 
ders. This ideal condition is represented diagram- 
matically in Fig. 25, where a large metal disk is shown 
at the top of the terminal. The object of this metal 
shield is to distribute the field between the terminal 
and the transformer cover in such a manner as to satisfy 
the above-mentioned condition. In practice, the ten- 
dency for corona to form at the exposed ends of the tin- 
foil cylinders is counteracted by treating the finished 
terminal with several coats of varnish, and surrounding 
it with an insulating cylinder filled with an insulating 
compound which can be poured in the liquid form and 
which solidifies at ordinary temperatures. This con- 
struction is shown in Fig. 26, which represents a prac- 
tical terminal of the condenser type. Compared with 
Fig. 24, it is longer, but appreciably smaller in diameter 
where it passes through the transformer cover. 

The dimensions of a condenser-type terminal such 
as illustrated in Fig. 26 may be determined approxi- 
mately as follows: Assuming the working pressure as 
88,000 volts, and the maximum permissible potential 
gradient in the dielectric (usually consisting of tightly 



Metal dim to control 
flax distribution. 

e of Iniultttng mtril. 

Iniu tlng compound. 

FIG. 26. Condenser-type Transformer Bushing Suitable for a Working 
Pressure of 88,000 Volts. 


wound layers of specially treated paper) as 90 kv.,* 
the maximum radial thickness of insulation required 

. , total volts 313 xv 

will be - -=^ = 3.48 cm. or (say) 1.5 

voltage gradient 90 

in. to include an ample allowance for the dividing 
layers of metal foil. If the inner tube is 2.25 in. in 
diameter, as in the previous example, the external 
diameter over the insulation at the center will be 2.25 
X3 = 5-25 in. instead of the 7.5 in. required for the 
previous design. 

It is customary to allow about 4000 volts per layer, 
and twenty-two layers of insulation alternating with 
twenty-two layers of tinfoil are used in this particular 
design. It is true that ideal conditions will not be 
actually fulfilled; the aggregate thickness of insulation 
might have to be slightly greater than 1.5 in., but the 
inner tube might be made 1.75 in. or 2 in. instead of 
2.25 in., and a practical terminal for 88,000- volt service 
could undoubtedly be constructed with a diameter over 
the insulation not exceeding 5.25 in. 

The projection of the terminal above the grounded 
plate (the cover of the transformer case) need not be 
so great as would be indicated by the application of the 
practical rule previously given for surface leakage dis- 

tance, namely, that this distance should be(o.5H -- -1 

\ 2 / 

in., where kv. stands for the working pressure. The 
reason why a somewhat shorter distance is permissible 
is that the surface of the terminal proper has been cov- 
ered by varnish and a solid compound, and so far as the 
enclosing cylinder is concerned, the stress along the sur- 
* Same as in the example of the compound-filled insulator. 


face of this cylinder will be fairly uniform, especially if 
a large flux-control shield is provided, as shown in Fig. 
26. In order to avoid the formation of corona at the 
lower terminal (below the surface of the oil) this end may 
conveniently be in the form of a sphere, the diameter 
of which would depend upon the voltage and the prox- 
imity of grounded metal. 

The following particulars relate to a condenser type 
bushing actually in service on 80,000 volts. The layers 
of insulation are built up on a metal tube of 2.25 in. 
outside diameter. The diameter over the outside in- 
sulating cylinder is 5.3 in. This bushing has uniform 
capacity, the thickness of the inner and outer insulating 
wall being the same, namely 0.062 in. ; but the thickness 
of the intermediate cylinders is variable, the maximum 
being 0.073 in. for the twelfth and thirteenth cylinders. 
(A plot of the individual thickness forms a hyperbolic 
curve.) The static shield or " hat " is 9 in. diameter and 
2 in. thick, the edge being rolled to a true semicircle. 
When provided with a casing filled with gum, and when 
the taper is such that the steps on the air end are 1.69 in. 
(total length = i.69X22=37.2 in.), there is no dif- 
ficulty in raising the voltage to 300,000 (r.m.s. value) 
without arc-over. The same bushing without a casing 
would arc-over at about 285,000 volts; but this can be 
raised to the same value as for the terminal with gum- 
filled casing if the size of the static shield is increased to 
about 2 "ft. diameter. When the arc-over voltage is 
reached, the discharge takes place between the edge of 
the v static shield and the flange which is bolted to the 
transformer case. 


20. Losses in Core and Windings. The power loss 
in the iron of the magnetic circuit is due partly to 
hysteresis and partly to eddy currents. The loss due 
to hysteresis is given approximately by the formula 

Watts per pound = KnB 1 ' 6 /, 

where K h is the hysteresis constant which depends upon 
the magnetic qualities of the iron. The symbols B and/ 
stand, respectively, for the maximum value of the mag- 
netic flux density, and the frequency. An approximate 
expression for the loss due to eddy .currents is 

Watts per pound = K e (Bft) 2 , 

where / is the thickness of the laminations, and K e is a 
constant which is proportional to the electric conduc- 
tivity of the iron. 

With the aid of such formulas, the hysteresis and eddy 
current losses may be calculated separately, and then 
added together to give the total watts lost per pound 
of the core material; but it is more convenient to use 
curves such as those of Fig. 27, which should be plotted 














from tests made on samples of the iron used in the con- 
struction of the transformer. These curves give the 
relation between maximum value of flux density, and 
total iron loss per pound at various frequencies. The 
curves of Fig. 27 are based on average values obtained 
with good samples of commercial transformer iron and 
silicon-steel; the thickness of the laminations being 
about 0.014 in. 

The cost of silicon-steel stampings is greater than that 
of ordinary transformer iron; but the smaller total iron 
loss resulting from the use of the former material will 
almost invariably lead to its adoption on economic 
grounds. The eddy-current losses are. smaller in the 
alloyed material than in iron laminations of the same 
thickness because of the higher electrical resistance of 
the former. The permeability of silicon-steel is slightly 
lower than that of ordinary iron, and this may lead to a 
somewhat larger magnetizing current; on the other hand, 
the modern alloyed transformer material (silicon-steel) is 
non-ageing, that is to say, it has not the disadvantage 
common to transformers constructed fifteen to twenty 
years ago, in which the iron losses increased appreciably 
during the first two or three years of operation. The 
" ageing " of the ordinary brands of transformer iron 
resulting in larger losses is caused by the material being 
maintained at a fairly high temperature for a consider- 
able length of time. 

The maximum flux density in transformer cores is 
generally kept below the knee of the B-H curve. As a 
guide for use in preliminary designs, usual values of B 
(gausses) are given below : 





/ = So or 60 

Small lighting or distributing 
Ordinary iron 
Alloyed iron 

8,000 to 11,000 
10,000 to 13,000 

5,000 to 7,000 
8,000 to 1 1 ooo 

Power transformers: 
Ordinary iron 

10,000 to 13,000 

9,000 to 11,000 

Alloyed iron 

11,000 to 14,000 

1 1, ooo to 14 ooo 

The losses in the iron core are usually less than one 
watt per pound, although they sometimes amount to 
1.5 watts, and even 1.8 watts, per pound. The higher 
figures apply to large, artificially cooled, power trans- 

Current Density in Windings. Even with well-ven- 
tilated coils (air blast), or improved methods of pro- 
ducing good oil circulation, the permissible current den- 
sity in the copper windings is limited by local heating. 
Jf the watts lost per pound of copper exceed a certain 
amount, there will be danger of internal temperatures 
sufficiently high to cause injury to the insulation. As a 
rough guide in deciding upon suitable values for trial 
dimensions in a preliminary design, the following approx- 
imate figures may be used : 


Type of Transformer. sSSTln?" 

Standard lighting transformers (oil -immersed; self-cooled). 800 to 1200 
Transformers for use in Central Generating Stations, or 

Substations (oil-cooled, or air blast) noo to 1600 

Large, carefully designed transformers, oil-insulated, with 

forced circulation of oil, or with water cooling-coils 1400 to 1900 


When the current is very large, it is important to sub- 
divide the conductors to prevent excessive loss by eddy 
currents. When flat strips are used, the laminations 
must be in the direction of the leakage flux lines. It is 
advisable to add from 10 to 15 per cent to the calculated 
I 2 R loss when the currents to be carried are large, even 
after reasonable precautions have been taken to avoid 
large local currents by subdividing the conductors. 

The mere subdivision of a conductor of large cross- 
section does not always eliminate the injurious effects 
of local currents in the copper, because, unless each of the 
several conductors that are joined in parallel at the ter- 
minals does not enclose the same amount of leakage 
flux, there will be different e.m.f.'s developed in various 
sections of the subdivided conductor, and consequent 
lack of uniformity in the current distribution. This ob- 
jection can sometimes be overcome by giving the assem- 
bled conductor (of many parallel wires or strips) a half 
twist, and so changing the position of the individual 
conductors relatively to the leakage flux; but, in any 
case, once this cause of increased copper loss is recog- 
nized, it is generally possible to dispose and join together 
the several elements of a compound conductor so that 
the leakage flux shall affect them all equally. 

21. Efficiency. The output of a single-phase trans- 
former, in watts, is 

W = EJ s cosd, 

where E s is the secondary terminal voltage; 7 S , -the sec- 
ondary current; and cos 0, the power factor of the 
secondary load. The percentage efficiency is then: 


100 X 

W + iron losses + copper losses 


All-day Efficiency. The all-day efficiency is a matter 
of importance in connection with distributing trans- 
formers, because, although the amount of the copper 
loss falls off rapidly as the load decreases, the iron loss 
continues usually during the twenty-four hours, and may 
be excessive in relation to the output when the trans- 
former is lightly loaded, or without any secondary load, 
during many hours in the day. 

What is understood by the all-day percentage efficiency 
is the ratio given below, the various items being cal- 
culated or estimated for a period of twenty-four hours: 

i ooX Secondary output in watt-hours 
Sec. watt-hrs.+watt-hrs. iron loss+watt-hrs. copper loss' 

It is in order that this quantity may be reasonably large 
that the iron losses in distributing transformers are 
usually less than in power transformers designed for the 
same maximum output. 

Efficiency of Modern Transformers. The alternating- 
current transformer is a very efficient piece of apparatus, 
as shown by the following figures which are an indication 
of what may be expected of well-designed transformers 
at the present time. 


Output, k.v.a. Efficiency (per cent) 

i From 94 . i to 96 

2 From 94 . 6 to 96 . 5 

5 From 95.5 to 97.3 

10 From 96 . 4 to 97 . 9 

20 From 97.2 to 98 . i 

50 From 97.6 to 98 . 4 


For a given cost of materials, the efficiency will improve 
with the higher frequencies, and a transformer designed 
for a frequency of 25 would rarely have an efficiency 
higher than the lower limit given in the above table, 
while the higher figures apply mainly to transformers 
for use on 6o-cycle circuits. 

The highest efficiency of a lighting transformer usually 
occurs at about three-quarters of full load. Typical 
figures for a 5 k.v.a. lighting transformer for use on a 
5o-cycle circuit are given below. 
Core loss = 46 watts. 
Copper loss (full load) = 114 watts. 
Calculated efficiency (100 per cent power factor) : 
At full load, 0.969. 
At three-quarters full load, 0.9713. 
At one-half full load, 0.9707. 
At one-quarter full load, 0.9583. 


(100 per cent power factor) 

Output, k.v.a. Efficiency, per cent. 

400 From 97.3 to 97 . 8 

800 From 97 . 7 to 98 . 2 

1 200 From 97.9 to 98 . 4 

2000 From 98 . i to 98 . 7 

2600 From 98 . 2 to 98 . 8 

The manner in which the efficiency of large power 
transformers falls off with increase of voltage (involving 
loss of space taken up by insulation) is indicated by the 


following figures, which refer to 1000 k.v.a. single-phase 
units designed for use on 5o-cycle circuits. 

H.T. Voltage. Fu " Load Effic ^ ncy 


Per cent. 
22,OOO 98.8 

33> 000 ---- ' 98.7 

44,000 98 . 5 

66,000 . 98.3 

88,000 , 98.0 

110,000 97.8 

The figures given below are actual test data showing 
the performance of some single-phase, oil-insulated, self- 
cooling, power transformers recently installed in a hydro- 
electric generating station in Canada: 

Output 400 k.v.a. 

Frequency /=6o 

Primary volts 2,200 

Secondary volts 22,000 

Core loss 1^760 watts 

Full-load copper loss 3, 550 watts 

Exciting current, 2.15 per cent, of full-load current. 
Temperature rise (by thermometer) after contin- 
uous full-load run, 36 C. 
Efficiency on unky power factor load : 

At 1.25 times full load.. . . 98.57 per cent 

At full load 98 . 7 

At three-quarter s full load . 98 . 7 5 

At one-half full load 98 . 65 

At one-quarter full load. . . 98 . o 


It should be stated that the core loss in these trans- 
formers was exceptionally low, being only 0.44 per cent of 
the k.v.a. output. The core losses in modern trans- 
formers will usually lie between the limits stated below: 

K.v.a. Output. 


Percentage Core Loss 
100 Xcore loss, watts. 

rated volt-ampere output' 

500. ... | 


0.75 to 0.95 

I . tO 1.2 



o . 6 to o . 7 

IOOO. j 


0.7 to i . o 



0.8 to 1. 15 



0.5 to 0.65 

2000 J. 


0.55 to 0.7 



0.7 to 0.95 

4000 | 


0.5 to o . 6 
0.6 to o. 75 

The core losses in small transformers for use on lighting 
circuits up to 2200 volts are usually less than i per cent 
for all sizes above 3 kw. They may be as low as 0.5 
per cent in a 50 kw. distributing transformer, and as high 
as 2.5 per cent in a i kw. transformer. 

The frequency, whether 25 or 60, does not greatly 
influence the customary allowance for core loss. 

Efficiency when Power Factor oj Load is Less than Unity. 
The total full-load losses (iron -f- copper) may be ex- 
pressed as a percentage of the k.v.a. output. Assume 
that these losses are equal to a(k.v.a.). Then at any 
power factor, cos 0. 


Efficiency = 

(k.v.a.) cos 0+a(k.v.a.) 

cos 0+a 

Let T\ stand for the efficiency at unity power factor, 


whence the efficiency at any power factor, cos 8, is 

cos 6 



?7 / 

As an example, calculate the full-load efficiency of a 
transformer on a load of 0.75 power factor, given that 
the efficiency on unity power factor is 0.969. 

The ratio of the total losses to the k.v.a. output is 


d= 2 ^ = 


whence the efficiency at 0.75 power factor is 


22. Temperature of Transformer Windings. Insu- 
lating materials such as cotton and paper, specially 
treated with insulating compounds or immersed in oil, 
may be subjected to a temperature up to, but not 
exceeding 105 C. The hottest spot of the winding 
cannot be reached by a thermometer, and it is therefore 
customary to add 15 C. to the temperature registered 
by a thermometer placed at the hottest accessible part 
of a transformer under test. The room temperature is 
frequently as high as 35 C. and the maximum permis- 
sible rise in temperature above that of -the surrounding 
air may be arrived at as follows : 

Permissible hottest spot temperature. 105 
Hottest spot correction 15 

Difference 90 

Assumed room temperature 35 

Difference ( = permissible temperature 
rise) 55 

Thus, under the worst conditions of heating, the per- 
missible temperature rise should not exceed 55 C. when 
the measurements are made with a thermometer. A 
more reliable means of arriving at transformer tempera- 
tures is to calculate these from resistance .measurements 
of the windings. Such measurements usually give some- 
what higher temperatures than when thermometers are 
used, and a hottest spot correction of 10 C. is then gen- 
erally recognized as sufficient. It should be noted, 


however, that room temperatures of 40 C. are not 
impossible, and it is therefore customary to limit the 
observed rise in temperature to 55 C. even when the 
resistance method of measuring temperatures is adopted. 
Transformers are usually designed to withstand an 
overload of two hours' duration after having been in 
continuous operation under normal full-load conditions. 
Either of the following methods of rating is to be found 
in modern transformer specifications: 

(1) The temperature rise not to exceed 40 C. on con- 
tinuous operation at normal load, and 55 C. after an 
additional two hours' run on 25 per cent overload. 

(2) The temperature rise not to exceed 35 C. on con- 
tinuous operation at normal load, and 55 C. after an 
additional two hours' run on 50 per cent overload. 

On account of the slow heating of the iron core, large 
oil-cooled transformers may require ten, or even twelve 
hours to attain the final temperature. 

23. Heat Conductivity of Insulating Materials. Be- 
fore discussing the means by which the heat is carried 
away from the external surface of the coils, it will be 
advisable to consider how the designer may predetermine 
approximately the difference in temperature between 
the hottest spot and the external surface of the windings. 
Calculations of internal temperatures cannot be made 
very accurately; but the nature of the problem is indi- 
cated by the following considerations: 

Fig. 28 is supposed to represent a section through a 
very large flat plate, of thickness t, consisting of any 
homogeneous material. Assume a difference of tem- 
perature of T d = (T-To)C. to be maintained between 


the two sides of the plate, and calculate the heat flow 
(expressed in watts) through a portion of the plate of 
area wXl. The resistance offered by the material of 
the plate to the passage of heat may be expressed in 
thermal ohms, the thermal ohm being denned as the 
thermal resistance which causes a drop of i C. per watt 


Watts =W 

FIG. 28. Diagram Illustrating Heat Flow through Flat Plate. 

of heat flow; or, if R h is the thermal resistance of the heat 
path under consideration, 


T" *- (I 


which permits of heat conduction problems being solved 
by methods of calculation similar to those used in con- 
nection with the electric circuit. 



Let k be the heat conductivity of the material, ex- 
pressed in watts per inch cube per degree Centigrade 
difference of temperature between opposite sides of the 

FIG. 29. Heat Conductivity: Heat Generated Inside Plate. 

cube, then the watts of heat flow crossing the area 
(wXl) square inches, as indicated in Fig. 28 is 


Fig. 29 illustrates a similar case, but the heat is now 
supposed to be generated in the mass of the material 
itself. We shall still consider the plate to be very large 


relatively to the thickness, so that the heat flow from 
the center outward will be in the direction of the hori- 
zontal dotted lines. A uniformly distributed electric 
current of density A amperes per square inch is supposed 
to be flowing to or from the observer, and the highest 
temperature will be on the plane YY' passing through 
the center of the plate. Assuming this plate to be of 
copper with a resistivity of o.84Xio~ 6 ohms per inch 
cube at a temperature of about 80 C., the watts lost in 
a section of area (xXw) sq. in. and length / in, will be 

W x = (Axw) 2 X 0.84X10- X 



By adapting Formula (20) to this particular case, the 
difference of temperature between the two sides of a 
section dx in. thick is seen to be 

dTd =w x x dx 


0.84 X A 2 A" , 
T<L = r , I xdx 

I O /v / 

0.84A 2 / 2 , 
= ^7-777 degrees Centigrade. . 

The value of k for copper is about 10 watts per inch 
cube per degree Centigrade. 

The problem of applying these principles to the prac- 
tical case of a transformer coil is complicated by the fact 



that the heat does not travel along parallel paths as in 
the preceding examples, and, further, that the thermal 
conductivity of the built-up coil depends upon the rel- 
ative thickness of copper and insulating materials, a 
relation which is usually different across the layers of 
winding from what it is in a direction parallel to the 


P, ,,. > m ,,,,,,,,,A 


!**(B o 








*'"L"y<' f y f y f yr< t 

f ^|f 

rr^' u 

FIG. 30. Diagram Illustrating Heat Paths in a Transformer Coil of 
Rectangular Cross-section. 

Fig. 30 represents a section through a transformer 
coil wound with layers of wire in the direction CM; 
the number of layers being such as to produce a total 
depth of winding equal to twice OB. The whole of the 
outside surface of this coil is supposed to be maintained 
at a constant temperature by the surrounding oil or air. 
In other words, it is assumed that there is a constant 
difference of temperature of T^ degrees between the 
hottest spot (supposed to be at the center 0) and any 
point on the surface of the coil. 


The heat generated in the mass of material is thought 
of as traveling outward through the walls of successive 
imaginary spaces of rectangular section and length / 
(measured perpendicularly to the plane of the section 
shown in Fig. 30), as indicated in the figure, where 
CDEF is the boundary of one of these imaginary spaces, 
the walls of which have a thickness dx in the direction 

OA , and a thickness dx ( } in the direction OB. 

(OB\ . 

(OA) " 

According to Formula (19), we can say that the dif- 
ference of temperature between the inner and outer 
boundaries of this imaginary wall is d 7^ = heat loss, in 
watts, occurring in the space CDEFXthe thermal 
resistance of the boundary walls. 

It is proposed to consider the heat flow through the 
portion of the boundary surface of which the area is 

If W x stands for the watts passing through this area, 
we can write 

2DElkg { 2CDlk h 



which simplifies into 


, , 

' ' (23) 


In order to calculate W x it is necessary to know not 
only the current density, A, but also the space factor, or 
ratio of copper cross-section to total cross- section. 

Let a stand for the thickness of copper per inch of total 
thickness of coil measured in the direction OA ; and let 
b stand- for a similar quantity measured in the direction 
OB ; the space factor is then (aXb), and 


Inserting this value of W x in (23), and making the 
necessary simplifications, we get 

dT d = 


whence, by integration between the limits x a and 

T d = r~ /TTTTTT de S- Cent - ( 2 4) 


Except for the obvious correction due to the intro- 
duction of the space factor (ab), the only difference 
between this formula and Formula (22) is that the 
thermal conductivity, instead of being k a , as it would 
be if the heat flow were in the direction OA only, is 


replaced by the quantity in brackets in the denominator 
of Formula (24). This quantity may be thought of as a 
fictitious thermal conductivity in the direction OA, 
which, being greater than k aj provides the necessary cor- 
rection due to the fact that heat is being conducted away 
in the direction OB, thus reducing the difference of tem- 
perature between the points and A. 

Calculation of k a and kb. 

Let k c and ki, respectively, stand for the thermal con- 
ductivity of copper and insulating materials as used in 
transformer construction. The numerical values of 
these quantities, expressed in watts per inch cube per 
degree Centigrade, are k c = 10 and k t = 0.0033. It follows 

10 ( 

, (25) 

<z_ (i a) a-\- 3000(1 a) 

and similarly, 



where a and b are the thickness of copper per inch of 
coil in the directions OA and OB, respectively, as pre- 
viously defined. 

Example. Suppose a transformer coil to be wound 
with 0.25X0.25 in. square copper wire insulated with 
cotton o.oi in. thick, and provided with extra insulation 
of 0.008 in. fullerboard between layers. There are 
twelve layers of wire and seven wires per layer. Assume 
the current density to be 1400 amperes per square inch, 

-- r~ , - ^rj 

2 XI06 [0.0448 +0.033 2 \ 


and calculate the hottest spot temperature if the outside 
surface of the coil is maintained at 75 C. 

0.25 0.25 

a = -. = 0.926; b = -5 = 0.9; whence space factor 

0.27 0.278 

(ab) =0.833. 

By Formulas (25) and (26), ^ = 0.0448, and ^ = 0.033 2; 
OA =3.5X0.27=0.945, and 0^ = 6X0.278 = 1.67 in. 

By Formula (24), 

=n Cent., 

and the hottest spot temperature = 75 + 11 =86. C. 

24. Cooling Transformers by Air Blast. Before the 
advantages of oil insulation had been realized, trans- 
formers were frequently enclosed in watertight cases, 
the metal of these cases being separated from the hot 
parts of the transformer by a layer of still air. This 
resulted either in high temperatures or in small kilowatts 
output per pound of material. Air insulation is still 
used in some designs of large transformers for pressures 
up to about 33,000 volts; but efficient cooling is ob- 
tained by forcing the air around the windings and 
through ducts provided not only between the coils, 
but also between the coils and core, and between sec- 
tions of the core itself. 

Since all the heat losses which are not radiated from 
the surface of the transformer case must be carried away 
by the air blast, it is a simple matter to calculate the 


weight (or volume) of air required to carry away these 
losses with a given average increase in temperature of 
outgoing over ingoing air. 

A cubic foot of air per minute, at ordinary atmospheric 
pressures, will carry away heat at the rate of about 0.6 
watt for every degree Centigrade increase of tempera- 
ture. Thus, if the difference of temperature between 
outgoing and ingoing air is 10 C., the quantity of air 
which must pass through the transformer for every kilo- 
watt of total loss that is not radiated from the surface 
of the case, is 

= = 1 66 cu. ft. per minute. 

v 0.6X10 

If the average increase in temperature of the air is 
from 10 to 15 C., the actual surface temperature rise of 
the windings may be from 40 to 50 C. ; the exact figure 
being difficult to calculate since it will depend upon the 
size and arrangement of the air ducts. The temperature 
of the coils is influenced not only by the velocity of the 
air over the heated surfaces, but also by the amount of 
the total air supply which comes into intimate contact 
with these surfaces. With air passages about J in. 
wide, and an average air velocity through the ducts 
ranging from 300 to 600 ft. per minute, the temperature 
rise of the coil surfaces will usually be from four to 
eight times the rise in temperature of the circulating 
air. Thus, although it is not possible to predetermine 
the exact quantity of air necessary to maintain the 


transformer windings at a safe temperature, this may be 
expressed approximately as: 

Cubic feet of air per minute 

r o /-i , Wt W r 

for 50 C. temperature rise = - --, 
r M r o.oX-%- 

of coil surface 

where W t total watts lost in transformer ; and 

W r = portion of total loss dissipated from surface 
of tank. 

The latter quantity may be estimated by assuming 
the temperature of the case to be about 10 C. higher 
than that of the surrounding air, and calculating the 
watts radiated from the case with the aid of the data in 
the succeeding article. 

Assuming W r to be 25 per cent of Wt, the Formula (27) 
indicates that about 150 cu. ft. of air per minute per 
kilowatt of total losses would be necessary to limit the 
temperature rise of the coils to 50 C. With poorly 
designed transformers, and also in the case of small 
units, the amount of air required may be appreciably 

It is true that, in turbo-generators, an allowance of 
100 cu. ft. per minute per kilowatt of total losses, is gen- 
erally sufficient to limit the temperature rise to about 
50 C.; but, owing to the churning of the air due to the 
rotation of the rotor, it would seem that the necessary 
supply of air is smaller for turbo-generators than for 


Filtered air is necessary in connection with air-blast 
cooling; otherwise the ventilating ducts are liable to 
become choked up with dirt, and high temperatures will 
result. Wet air niters are very satisfactory and desir- 
able, provided the amount of moisture in the air passing 
through the transformers is not sufficient to cause a 
deposit of water particles on the coils. Air containing 
from i to 3 per cent of free water in suspension is a much 
more effective cooling medium than dry air. It would 
probably be inadvisable to use anything but dry air in 
contact with extra-high voltage apparatus; but trans- 
formers for very high pressures are not designed for 
air-blast cooling.* 

25. Oil-immersed Transformers Self Cooling. The 
natural circulation of the oil as it rises from the heated 
surfaces of the core and windings, and flows do w a ward 
near the sides of the containing tank, wil-l lead to a tem- 
perature distribution generally as indicated in Fig. 31. 
The temperature of the oil at the hottest part (close to 
the windings at the top of the transformer) will be some- 
what higher than the maximum temperature of the tank, 
which, however, will be hotter in the neighborhood of the 
oil level than at other parts of its surface. The average 
temperature of the cooling surface in contact with the 
air bears some relation to the highest oil temperature, 
and, since this relation does not vary greatly with 
different designs of transformer, or case, a curve such 

* Some useful data on the relative cooling effects of moist and dry air, 
together with test figures relating to a i2-kw. air-cooled transformer, 
will be found in Mr. F. J. Teago's paper " Experiments on Air-blast 
Cooling of Transformers," in the Jour. Inst. E. ., May i, 1914 ,Vol. 52, 
page 563. 



as Fig. 32 may be used for calculating the approximate 
tank are a necessary to prevent excessive oil temperatures. 
The oil temperature referred to in Fig. 32 is the dif- 
ference in degrees Centigrade between the temperature 
of the hottest part of the oil and the air outside the tank. 

Temperature of < 

FIG. 31. Distribution of Temperature with Transformer Immersed in 


This will be somewhat greater than the temperature rise 
of any portion of the transformer case; but the curve 
indicates the (approximate) number of watts that can 
be dissipated by radiation and air currents per square 
inch of tank surface. The curve is based on average 
figures obtained from tests on tanks with smooth surfaces 














0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 

'4= > Watt8 dissipated per sq. in. of tank surface 

FIG. 32. Curve for Calculating Cooling Area of Transformer Tanks. 


(not corrugated), the surface considered being the total 
area of the (vertical) sides plus one-half the area of the 
lid. The cooling effect of the bottom of the tank 'is 
practically negligible, and is not to be included in the 

Example. What will be the probable maximum tem- 
perature rise of the oil in a self-cooling transformer with 
a total loss of 1200 watts, the tank of sheet-iron with- 
out corrugations measuring 2 ft.X2 ft.X3-5 ft. high? 

The surface for use in the calculations is S = (3. 5X8) 
+2 = 30 sq. ft., whence 

1 200 
w = =0.278, 


which', according to Fig. 32, indicates a 43 C. rise of 
temperature for the oil. 

The temperature of the windings at the hottest part of 
the surface in contact with the oil might be from 5 to 10 
C. higher than the maximum oil temperature as meas- 
ured by thermometer. Assume this to be 7 C. Assume 
also that the room temperature is 35 C. and that the 
difference of temperature (To) between the coil surface 
and the hottest spot of the windings as calculated by 
the method explained in Art. 23 is 13 C. Then the 
hottest spot temperature in the transformer under con- 
sideration would be about 35+43 + 7 + 13 = 98 C. 

26. Effect of Corrugations in Vertical Sides of Con- 
taining Tank. The cooling surface in contact with the 
air may be increased by using corrugated sheet-iron 
tanks in place of tanks with smooth sides. It must not, 


however, be supposed that the temperature reduction 
will be proportional to the increase of tank surface 
provided in this manner; the watts radiated per square 
inch of surface of a tank with corrugated sides will always 
be appreciably less than when the tank has smooth sides. 
Not only is the surface near the bottom of the corruga- 
tions less effective in radiating heat than the outside 
portions; but the depth and pitch of the corrugations 
will affect the (downward) rate of flow of the oil on the 
inside of the tank, and the (upward) convection cur- 
rents of air on the outside. 

It is practically impossible to develop formulas which 
will take accurate account of all the factors involved, and 
recourse must therefore be had to empirical formulas 
based on available test data together with such reason- 
able assumptions as may be necessary to render them 
suitable for general application. 

If X is the pitch of the corrugations, measured on the 
outside of the tank, and / is the surface width of material 
per pitch (see the sketch in Fig. 33), the ratio of the 
actual tank surface to the surface of a tank without 

corrugations is -. The heat dissipation will not be 


in this proportion because, although the cooling effect 
will increase as / is made larger relatively to X, the 
additional surface becomes less and less effective in 
radiating heat as the depth of the corrugations increases 
without a corresponding increase in the pitch. It is 
convenient to think of the surface of an equivalent 
smooth tank which will give the same temperature 
rise of the oil as will be obtained with the actual tank. 



If we apply a correction to the actual pitch, X, 
and obtain an equivalent pitch, \ e , the ratio k = 


is a factor by which the tank surface (neglecting corru- 














FIG. 33. Curve Giving Factor k for Calculating Equivalent Cooling 
Surface of Tanks with Corrugated Sides. 

gations) must be multiplied in order to obtain the 
equivalent or effective surface. If all portions of the 
added surface were equally effective in radiating heat, 
no correcting factor would be required, and the equiva- 
lent pitch would be obtained by adding to X the quan- 


tity (/ X); but since a modifying factor is needed, 
the writer proposes the formula 

, .... (28) 

wherein the additional surface provided by the cor- 


rugations is reduced in the ratio - which becomes 


unity when / = X. A modifying factor of this form not 
only seems reasonable on theoretical grounds, but it 
is required in an empirical formula based on available 
experimental data. It follows that 

't x 

or, if- 

Values of &, as obtained from this formula for different 
values of n, may be read off the curve Fig. 33. 

Example. What would have been the temperature 
rise of the oil if, instead of the smooth-side tank of the 
preceding example (Art. 25), a tank of the same external 
dimensions had been provided with corrugations 2 in. 
deep, spaced ij in. apart? 

The approximate value of / is 1.25+4 = 5.25 in. 

Whence w = ? -=0.238; and from the curve, Fig. 30, 
= 2.23. 


The equivalent tank surface is 5 = (3.5X8X2. 23) + 2 
= 64.5 sq. in., whence 

w = - = 0.129, 


which, according to Fig. 32, indicates a 27 C. rise of 
temperature, as compared with 43 C. with the smooth- 
surface tank of the same outside dimensions. 

27. Effect of Overloads on Transformer Tempera- 
tures. Since the curve of Fig. 32 is not a straight line, 
it follows that the watts dissipated per square inch of 
tank surface are not directly proportional to the differ- 
ence between the oil, and room, temperatures. The 
approximate relation, according to this curve, is 

Temperature rise = constant X w' 6 , . . (31) 

which may be used for calculating the temperature rise 
of a self-cooling oil-immersed transformer when the tem- 
perature rise under given conditions of loading is known. 
Example. Given the following particulars relating to 
a transformer'. 

Core loss = 100 watts, 

Copper loss (full load) = 200, 

Final temp, rise (full load) of the 011 = 35 C. 

Calculate the final temperature rise after a continuous 
run at 20 per cent overload. 

For an increase of 20 per cent in the load, the copper 


loss is 2ooX (i. 2)2 = 288 watts; whence, according to 
Formula (31): 

- 6 

/2SS-hIOoV' u o r^ 

Temperature rise = 35X1- - I =41 C. approx. 


The calculation of temperature rise resulting from an 
overload of short duration is not so simple. It is neces- 
sary to take account of the specific heat of the materials, 
especially the oil, because the heat units absorbed by 
the materials have not to be radiated from the tank 
surface, and the calculated temperature rise would be 
too high if this item were neglected. 

The specific heat of a substance is the number of calor- 
ies required to raise the temperature of i gram i C., 
the specific heat of water being taken as unity. 

The specific heat of copper is 0.093, an d for an average 
quality of transformer oil, it is 0.32. 

One gram-calorie (i.e., the heat necessary to raise the 
temperature of i gram of water i C.)=4.i83 joules (or 
watt-seconds). Also, i lb.=453.6 grams. It follows 
that the amount of energy in watt-seconds necessary to 
raise M c pounds of copper T C. is 

Watts X time in seconds = 4. 183X0.093X453. 6 M C T 
= 177 M C T (for copper). 

Similarly, if we put M for the weight of oil, in pounds, 
and replace the figure 0.093 by 0.32, we get 

Watts X time in seconds = 6 10 M T (for oil). 


In the case of an overload after the transformer has 
been operating a considerable length of time on normal 
full load, all the additional losses occur in the copper 
coils, and it is generally permissible to neglect the 
heat absorption by the iron core. We shall, therefore, 
assume that the additional heat units which are not 
absorbed by the copper pass into the oil, and that the 
balance, which is not needed to heat up the oil, must be 
dissipated by radiation and convection from the sides 
of the containing tank. It will greatly simplify the cal- 
culations if we further assume that the watts dissipated 
per square, inch of tank surface per degree difference of 
temperature are constant over the range of temperature 
involved in the problem. (By estimating the average 
temperature rise, and finding w on the curve, Fig. 32, 


a suitable value for the quantity may be selected.) 

If W t = total watts lost (iron -[-copper), the total 
energy loss in the interval of time dt second 
is Wtdt. 

If the increase of temperature during this interval of 
time is dx degree Centigrade, the heat units absorbed 
by the copper coils and the oil are K s dx, where K s = 

The difference between these two quantities represents 
the number of joules, or watt-seconds, of energy to be 

* In order to simplify the calculations, it has been (incorrectly) assumed 
that the temperature rise of the copper is the same as that of the oil. 
It will, of course, be somewhat greater; but since the heat absorbed by 
the copper is small compared with that absorbed by the oil, this assump- 
tion will not lead to an error of appreciable magnitude. 


radiated from the tank surface during the interval of 
time dt second; whence, 

, .... (32) 

where Kr = t&nk surface in square inches X radiation 
coefficient, in watts per square inch per i C. rise, and 
# = the initial oil temperature rise (which has been 
increased, by the amount dx) . 

Equation (32) may be put in the form 

dL = Ks 
dx W t 

The limits for x are the initial oil temperature TQ 
and the final oil temperature T t , which is reached at 
the end of the time /. Therefore, 


If time is expressed in minutes, and common logs, 
are used, we have, 

w, T \ 

K s , lK~r~ 10 \ . , . 

- minutes. . . (34) 

\ Wt T I 

\K- Tl / 


In order to facilitate the use of this formula, the 
meaning of the symbols is repeated below: 

Wt total watts lost (iron + copper) ; 

K T =SX radiation coefficient expressed in watts per 

square inch per i C. rise of temperature of the 

where S = tank surface in square inches, as de- 

fined in Art. 25, corrected if necessary for 

corrugations (Art. 26). 

where M c = weight of copper (pounds) ; 

and MQ = weight of oil (pounds) ; 
TO = initial temperature of oil (degrees C.); 
T t = temperature of oil (degrees C.) after the overload 

(producing the total losses Wt) has been on for t m 


Example. Using the data of the preceding example, 
the full-load conditions are: 

Core loss = 100 watts; 

Copper loss = 200 watts; 

Temperature rise = 3 5 C. 

Referring to Fig. 29, the value for w for a temper- 
ature rise of 35 is 0.193, from which it follows that 

the effective tank surface is S = ~ = 1550 sq. in. 

* yo 
Given the additional data: 

Weight of copper = 65 lb., 
Weight of oil = 140 lb. 


calculate the time required to raise the oil from T = 35 C. 
to r* = 45 C. on an overload of 50 per cent. 
The copper loss is now 200 X (i-5) 2 =450 watts, whence 

Wt 100+450 = 550 watts. 

The cooling coefficient (from curve, Fig. 32), for an 
average temperature rise of - -=40 C., is - 
= 0.00606, whence, 

K r = 

# s =(l77X65) + (610X140) =97,000; 

and, by Formula (34), 


28. Self-cooling Transformers for Large Outputs. 

The best way to cool large transformers is to provide 
them with pipe coils through which cold water is circu- 
lated, or, alternatively, to force the oil through the ducts 
and provide means for cooling the circulating oil outside 
the transformer case. When such methods cannot be 
adopted as in most outdoor installations and other 
sub-stations without the necessary machinery and at- 
tendants the heat from self-cooling transformers of 
large size is dissipated by providing additional cooling 
surface in the form of tubes, or flat tanks of small volume 



and large external surface, connected to the outside of a 
central containing tank. Unless test data are available 
in connection with the particular design adopted, judg- 
ment is needed to determine the effective cooling surface 
(see Art. 26) in order that the curve of Fig. 32 or such 
cooling data as may be available for smooth-surface 
tanks may be used for calculating the probable tem- 
perature rise. 

In the tubular type of transformer tank which is pro- 
vided with external vertical tubes connecting the bottom 

of the tank to the level, 
near the oil surface, where 
the temperature is highest 
(as roughly illustrated by 
Fig. 34), the tubes should 
be of fairly large diameter 
with sufficient distance be- 
tween them to allow free 
circulation of the air and 
efficient radiation. It is 
not economical to use a 
very large number of small 
tubes closely spaced with 
a view to obtaining a large 
cooling surface, because the 

FIG. 34. Transformer Case with r . , . , , 

Tubes to Provide Additional Cool- CXtra SUrfaCC obtained by 

ing Surface. such means is not as 

effective as when wider 

spacing is used. If the added pipe surface, A p , is 1.5 
times the tank surface, A tj without the pipes, the effective 
cooling surface will be about S = (A t +A p )Xo.g] but, 


with a greatly increased surface obtained by reducing 
the spacing between the pipes, the correction factor 
might be very much smaller than 0.9. 

29. Water-cooled Transformers. The cooling coil 
should be constructed preferably of seamless copper tube 
about ij in. diameter, placed near the top of the tank, 
but below the surface of the oil. If water is passed 
through the coil, heat will be carried away at the rate of 
1000 watts for every 3! gals, flowing per minute when the 
difference of temperature between the outgoing and 
ingoing water is i C. Allowing 0.25 gal. per minute, 
per kilowatt, the average temperature rise of the water 

will be = 15 C. The temperature rise of the oil is 

considerably greater than this: it will depend upon the 
area of the coil in contact with the oil and the condition 
of the inside surface, which may become coated with 
scale. An allowance of i sq. in. of coil surface per watt 
is customary; but the rate at which heat is transferred 
from the oil to the water may be from 2 to 2j times as 
great when the pipes are new than after they have 
become coated with scale. It may, therefore, be neces- 
sary to clean them out with acid at regular intervals, if 
the danger of high oil temperatures is to be avoided. 

Example. Calculate the coil surface and the quan- 
tity of water required for a transformer with total losses 
amounting to 6 kw., of which it is estimated that 2 
kw. will be dissipated from the outside of the tank. 
Surface of cooling coil = 6000 2000 = 4000 sq. in. 

Assuming a diameter of ij in., the length of tube in 


the coil will have to be -7 = 85 ft. 



The approximate quantity of water required will be 
0.25X4 = 1 gal. per minute. 

30. Transformers Cooled by Forced Oil Circulation. 
The transformer and case are specially designed so that 
the oil may be forced (by means of an external pump) 
through the spaces provided between the coils and be- 
tween the sections of the iron core. The ducts may 
be narrower than when the cooling is by natural circula- 
tion of the oil. The capacity of the oil pump may be 
estimated by allowing a rate of flow of oil through the 
ducts ranging from 20 to 30 ft. per minute. 

It is not essential that the oil be cooled outside the 
transformer case; in some modern transformers, the con- 
taining tank proper is surrounded by an outer case, and 
the space between these two shells contains the cooling 
coils through which water is circulated. These coils, 
instead of being confined to the upper portion of the 
transformer case, as when water cooling is used without 
forced oil circulation, may occupy the whole of the space 
between the inner and outer shells of the containing tank. 
The oil circulation is obtained by forcing the oil up 
through the inner chamber and downward in the space 
surrounding the water cooling-coils. 

Such systems of artificial circulation of both oil and 
water are very effective in connection with units of large 
output; but they could not be applied economically to 
medium-sized or small units. 



31. Magnetic Leakage. Assuming the voltage applied 
to the terminals of a transformer to remain constant, it 
follows that the flux linkages necessary to produce the 
required back e.m.f. can readily be calculated. The 
(vectorial) difference between the applied volts and the 
induced volts must always be exactly equal to the ohmic 
drop of pressure in the primary winding. Thus, the 
total primary flux linkages (which may include leakage 
lines) must be such as to induce a back e.m.f. very nearly 
equal to the applied e.m.f. the primary IR drop being 
comparatively small. 

When the secondary is open-circuited, practically all 
the flux linking with the primary turns links also with 
the secondary turns; but when the transformer is loaded, 
the m.m.f. due to the current in the secondary winding 
has a tendency to modify the flux distribution, the action 
being briefly as follows: 

The magnetomotive force due to a current I s flowing 
in the secondary coils would have an immediate effect 
on the flux in the iron core if it were not for the fact that 
the slightest tendency to change the number of flux lines 
through the primary coils instantly causes the primary 
current to rise to a value I p such that the resultant 



ampere turns (I P T P I S T S ) will produce the exact amount 
of flux required to develop the necessary back e.m.f. 
in the primary winding. Thus, the total amount of 
flux linking with the primary turns will not change 
appreciably when current is drawn from the secondary 
terminals; but the secondary m.m.f. together with an 
exactly ^qual but opposite primary magnetizing effect 
will cause some of the flux which previously passed 
through the secondary core to " spill over " and avoid 
some, or all, of the secondary turns. This reduces the 
secondary volts by an amount exceeding what can be 
accounted for by the ohmic resistance of the windings. 
Although it is possible to think of a leakage field set 
up by the secondary ampere turns independently of that 
set up by the primary ampere turns, these imaginary flux 
components must be superimposed on the main flux 
common to both primary and secondary in order that 
the resultant magnetic flux distribution under load may 
be realized. The leakage flux is caused by trie com- 
bined action of primary and secondary ampere turns, 
and it is incorrect, and sometimes misleading, to think 
of the secondary leakage reactance of a transformer as if 
it were distinct from primary reactance, and due to 
a particular set of flux lines created by the secondary 
current. In order to obtain a physical conception of 
magnetic leakage in transformers it is much better to 
assume that the secondary of an ordinary transformer 
has no ^//-inductance, and that the loss of pressure 
(other than IR drop) which occurs under load is caused 
by the secondary ampere turns diverting a certain amount 
of magnetic flux which, although it still links with the 


primary turns, now follows certain leakage paths instead 
of passing through the core under the secondary coils. 

32. Effect of Magnetic Leakage on Voltage Regula- 
tion. The regulation of a transformer may be defined 
as the percentage increase of secondary terminal voltage 
when the load is disconnected (primary impressed volt- 
age and frequency remaining unaltered). 

The connection between magnetic leakage and voltage 
regulation will be studied by considering the simplest 
possible cases, and noting the difference in secondary 
flux-linkages under loaded and open-circuited conditions. 
The amount of the leakage flux in proportion to the 
useful flux will purposely be greatly exaggerated, and, 
in order to eliminate unessential considerations, the fol- 
lowing assumptions will be made : 

(1) The magnetizing component of the primary cur- 
rent will be considered negligible relatively to the total 
current, and will not be shown in the diagrams. 

(2) The voltage drop due to ohmic resistance of both 
primary and secondary windings will be neglected. 

(3) The primary and secondary windings will be sym- 
metrically placed and will consist of the same number of 

(4) One flux line as shown in the diagrams linking 
with one turn of winding will generate one volt. 

In Fig. 35, both primary and secondary coils consist 
of one turn of wire wound close around the core: a cur- 
rent 7 S is drawn from the secondary on a load of power 
factor cos 0, causing a current /i exactly equal but 
opposite to Is to flow in the primary coil, the result being 
the leakage flux as represented by the four dotted lines. 



The secondary voltage, E s = 2 volts, is due to the two 
flux lines which link both with the primary and secondary 


FIG. 35. Magnetic Leakage: Thickness of Coils Considered Negligible. 

coils. The phase of this component of the total flux is, 
therefore, 90 in advance of E s as indicated by the line 
OB in the vector diagram. 


In order to calculate the necessary primary impressed 
e.m.f., we have as one component OE'\ exactly equal but 
opposite to OE S because the flux OB will induce in the 
primary coil a voltage exactly equal to E s in the second- 
ary. The other component is E f \E v = ^. volts, equal but 
opposite to the counter e.m.f. which, being due to the 
four leakage lines created by the current /i, will lag 90 
in phase behind OI\. The resultant is OE P which scales 
5 volts.* 

When load is thrown off the transformer there will be 
five lines linking with the primary which, since there is 
now no secondary m.m.f. to produce leakage flux, will 
pass through the iron core and link with the secondary. 
The secondary voltage on open circuit will, therefore, 
be E p = 5 volts, and the percentage regulation is 

EpE s 5 2 
looX ^ = iooX = 150. 

<<! 2 

In Fig. 36, a departure is made from the extreme 
simplicity of the preceding case in order to illustrate the 
effect of leakage lines passing not only entirely outside 
the windings, but also through the thickness of the coils, 
as must always happen in practical transformers where 
the coils occupy an appreciable amount of space. 

Each winding now consists of two turns, with an air 
space between the turns through which leakage flux 

* The reason why the six flux lines shown in the figure as linking 
with the primary coil do not generate 6 volts is, of course, due to the 
fact that these flux lines are not all in the same phase; the resultant 
or actual flux in the core under the primary coil is 5 lines, as indicated 
by the vector diagram. The actual amount of flux passing any given 
cross-section of the core must be thought of as the (vectorial) addition 
of the flux lines shown in the sketch at that particular section. 



represented in Fig. 36 by one dotted line is supposed to 
pass. The single flux line, linking with both the primary 








::rr- v > 

I t t 
s> i @ 


s \ 

l v ~: . _' 


FIG. 36. Magnetic Leakage: Thickness of Coils Appreciable. 

and secondary windings, generates the e.m.f. component 
E = 2 volts. The leakage flux line marked F links with 


only one turn of the secondary, and therefore generates 
one volt lagging 90 in phase behind the primary current 
/i. The total secondary voltage is E s which scales 2.6 
volts; the balancing component in the primary being 
EI. It should be particularly noted that this balancing 
component does not account for the full effect of the two 
flux lines B and F linking with the primary, because, 
while the flux line F links with only one secondary turn, 
it links with two primary turns. The voltage com- 
ponent OE'i in the primary may, therefore, be thought 
of as due to the flux lines B and /, leaving for the remain- 
ing component of the impressed e.m.f., E\E P = 6 volts 
(leading O/i by 90) which may be considered as caused 
by the three lines F, H, and G. In other words, the 
reactive drop (I\X P ) depends upon the di/erence between 
the primary and secondary flux-linkages of the stray 
magnetic field set up by the combined action of the 
secondary current I s and the balancing component /i 
of the total primary current. (In this case /i is the 
total primary current, since the magnetizing component 
is neglected.) The leakage flux-linkages are as follows: 
With the primary turns : 

JXi = i volt, 
HX2 = 2 volts, 
GX2 = 2 volts, 
FX2 = 2 volts 

7 volts 
With the secondary turns: 

FXi = i volt 

giving a difference of 6 volts 


This is the. vector (I\X P }. When applying this rule 

to actual transformers in which the ratio of turns ~ is 

1 s 

not unity, the proper correction must be made (as ex- 
plained later) when calculating the equivalent e.m.f. 
component in the primary circuit. 

To obtain the regulation in the case of Fig. 36, we 
have E P = S volts and E s = 2.6 volts when the transformer 
is loaded. When the load is thrown off, there will be 
four flux lines linking with both primary and secondary 
producing 8 volts in each winding. The regulation is 


100 X ^- = 208 per cent. 

33. Experimental Determination of the Leakage Reac- 
tance of a Transformer. Although these articles are 
written from the viewpoint of the designer, who must 
predetermine the performance of the apparatus he is 
designing, a useful purpose will be served by considering 
how the leakage reactance of an actual transformer may 
be determined on test. The purpose referred to is the 
clearing up of any vagueness and consequent inaccuracy 
that may exist in the mind of the reader, due largely 
in the writer's opinion to the common, but unnecessary 
if not misleading assumption, that the secondary has 
self induction.* 

* The assumption usually made in text books is that the secondary 
self-induction (i.e., the flux produced by the secondary current, and 
linking with the secondary turns) is equal to the primary leakage 


The diagram, Fig. 37, shows the secondary of a 
transformer short-circuited through an ammeter, A, 
of negligible resistance. The impressed primary voltage 
E z , of the frequency for which the transformer is de- 
signed, is adjusted until the secondary current I s is 
indicated by the ammeter. If the number of turns 
in the primary and secondary are T p and T s respectively, 

IT \ 
the primary current will be 7i=/ s ( M because, the 

\7 p/ 
amount of flux in the core being very small, the mag- 

FIG. 37. Diagram of Short-circuited Transformer. 

netizing component of the primary current may be 

The measured resistances RI and R 2 of the primary 
and secondary coils being known, the vector diagram 
Fig. 38, can be constructed. 

The volts induced in the secondary are OE% (equal 
to I S R2) in phase with the current I s . The bal- 
ancing component in the .primary winding is OE'\ 

equal to 2 ( ~ j in phase with the primary current I\. 

Another component in phase with this current is E'\P 
(equal to I\R\). Since the total impressed voltage 



has the known value E z , we can describe an arc of 
circle of radius OE Z from the point as a center. By 
erecting a perpendicular to O/i at the point P, the 
point E z is determined, and E Z P is the loss of pressure 
caused by magnetic leakage. The vector OP may be 
thought of as the product of the primary current /i, 

FIG. 38. Vector Diagram of Short-circuited Transformer. 

and an equivalent primary resistance R p , which assumes 
the secondary resistance to be zero, but the primary 
resistance to be increased by an amount equivalent 
to the actual secondary resistance. Thus, 






...... (35) 

In order to get an expression for the transformer 
leakage reactance (X p ) in terms of the test data, we 
can write, 


This quantity, multiplied by /i (or IiX p = 
VE z 2 -(IiR P ) 2 ) is the vector E'iE p of the diagrams 
in Figs. 35 and 36 as it might be determined experi- 
mentally for an actual transformer. If it were possible 
for all the magnetic flux to link with all the primary, 
and all the secondary, turns, the quantity IiX P would 
necessarily be zero; all the flux would be in the phase 
OB, and OE Z (of Fig. 38) would be equal to OP. The 
presence of the quantity IiX P can only be due to those 
flux lines which link with primary turns, but do not link 
with an equivalent number of secondary turns. 

34. Calculation of Reactive Voltage Drop. Seeing that 
it is generally although not always desirable to obtain 
good regulation in transformers, it is obvious that designs 
with the primary and secondary windings on separate 
cores (see Figs, i, 35 and 36), which greatly exaggerate 
the ratio of leakage flux to useful flux, would be very 
unsatisfactory in practice. By putting half the primary 
and half the secondary on each of the two limbs of a 



single-phase core-type transformer, as shown in Fig. 39, 
a considerable improvement is effected, but the reluctance 
of the leakage paths is still low, and this design is not 
nearly so good as Fig. 7 (page 18) where the leakage paths 
have a greater length in proportion to the cross-section. 
Similarly in the shell type of transformer, the design 
shown in Fig. 40 is unsatisfactory; the arrangement of 

FIG. 39. Leakage Flux Lines in Special Core-type Transformer. 

coils, as shown in Figs. 10 and n (Art. 8) is much better 
because of the greater reluctance of the leakage paths. 
Transformers with coils arranged as in Figs. 7 and 10 are 
satisfactory for small sizes; but, in large units, it is neces- 
sary to subdivide the windings into a large number of 
sections with primary coils " sandwiched " between 
secondary coils as in Fig. 17 (core type) and Figs. 8. and 



1 6 (shell type). By subdividing the windings in this 
manner, the m.m.f. producing the leakage flux, and the 
number of turns which this flux links with, are both 
greatly reduced. The objection to a very large number 
of sections is the extra space taken up by insulation 
between the primary and secondary coils. For the 
purpose of facilitating calculations, the windings of 
transformers can generally be divided into unit sections 

FIG. 40. Leakage Flux Lines in Poorly Designed Shell-type Transformer. 

as indicated in Fig. 41 (which shows an arrangement of 
coils in a shell- type transformer similar to Fig. 16). 
Each section consists of half a primary coil and half a 
secondary coil, with leakage flux passing through the 
coils and the insulation between them * all in the same 

* If air ducts are required between sections of the winding, these should 
be provided in the position of the dotted center lines, by a further sub- 
division of each primary and secondary group of turns; thus allowing 
the space between primary and secondary coils to be filled with solid 
insulation. It is evident that, if good regulation is desired, the space 
between primary and secondary coils where the leakage flux density 
has its maximum value must be kept as small as possible. 


direction, as indicated by the flux diagram at the bottom 
of the figure. 

Unit section.. 


FIG. 41. Section through Coils of Shell-type Transformer. 

The effect of all leakage lines in the gap between the 
coils is to produce a back e.m.f. in the primary without 


affecting the voltage induced in the secondary by the 
main component of the total flux (represented by the 
full line). Of the other leakage lines, B links with only 
a portion of the primary turns and has no effect on the 
primary turns which it does not link with ; while A links 
not only with all the primary turns, but also with a cer- 
tain number of secondary turns. Note that if the line A 
were to coincide with the dotted center line MN, 
marking the limit of the unit section under consideration, 
it would have no effect on the transformer regulation 
because flux which links equally with primary and 
secondary is not leakage flux. Actually, the line A 
links with all the primary turns of the half coil in the 
section considered, but with only a portion of the second- 
ary turns in the same section, Its effect is, therefore, 
exactly as if it linked with only a fractional number of 
the primary turns. The mathematical .development 
which follows is based on these considerations. 

Fig. 42 is an enlarged view of the unit section of Fig. 41, 
the length of which measured perpendicularly to the 
cross section -is / cms.- All the leakage is supposed to 
be along parallel lines perpendicular to the surface of the 
iron core above and below the coils. 

It is desired to calculate the reactive voltage drop in a 
section of the winding of length / cms., depth h cms., and 
total width (s+g+p) cms., where 

s = the half thickness of the secondary coil; 

g = the thickness of insulation between primary and 

secondary coils; 
p = the half thickness of the primary coil. 



The voltage drop caused by the leakage flux in the 
spaces g, p, and s will be calculated separately and then 
added together to obtain the total reactive voltage drop. 
The general formula giving the r.m.s. value of the volts 
induced by < maxwells linking with T turns is 


when the flux variation follows the simple harmonic law. 

FIG. 42. Enlarged Section through Transformer Coils. 

In calculating the voltage produced by a portion of 
flux in a given path, we must therefore determine (i) 


the amount of this flux, and (2) the number of turns 
with which it links. The symbols T\ and TI will be 
used to denote the number of turns in the half sections 
of widths p and 5 of the primary and secondary coils 
respectively. The meaning of the variables x and y 
is indicated in Fig. 42. The symbol m will be used for 


the quantity -^. 
For the section g we have, 

Inserting for 3> its value in terms of m.m.f. and per- 
meance, this becomes," 

(7^ = ^(04^70 X^XTY . . (37) 

In the section p, the m.m.f. producing the element 
of flux in the space of width dx is due to the current 

7i in (- }Ti turns, and since this element of flux links 

. /*\ 
in ( - 


- ) T\ turns, we have, 





In the section s, the m.m.f., producing the small element 
of flux in the space of width dy is due to the current 

7, in (-}TI turns, and since this must be considered 
as linking with ( i ) T\ turns, we can write, 


s 2 h 

*/ \j 

i- (39) 

wherefrom the -secondary quantities T 2 and I s have been 
eliminated by putting (Ti/i) in place of (TWs). 

The final expression for the inductive voltage drop 
in the unit section considered is obtained by adding 
together the quantities (37), (38), and (39). Thus, 

' (40) 

wherein all dimensions are expressed in centimeters. 
If all the primary turns are connected in series, this 


quantity will have to be multipled by the ratio - (or 


by twice the number of primary groups of coils) to 
obtain the value of the vector I\ X p shown in the vector 


Equivalent Value of the Length I. The numerical value 
of the length / as used in the above formulas might 
reasonably be taken as the mean length per turn of the 
transformer windings, provided the reluctance of the 
flux paths outside the section shown in Fig. 42 may be 
neglected, not only where the iron laminations provide 
an easy path for the flux, but also where the ends of the 
coils project beyond the stampings. 

Every manufacturer of transformers who has accu- 
mulated sufficient test data from transformers built 
to his particular designs, will be in a position to modify 
Formula (40) in order that it may accord very closely 
with the measured reactive voltage drop. This correc- 
tion may be in the form of an expression for the equiva- 
lent length /, which takes into account the type of 
transformer (whether core or shell) and the arrange- 
ment of coils; or the quantity g + may be 

modified, being perhaps more nearly g+ , which 

allows for more leakage flux through the space occupied 
by the copper than is accounted for on the assumption 
of parallel flux lines. The writer believes, however, 
that if / is taken equal to the mean length per turn 
of the windings expressed in centimeters the Formula 
(40) will yield results sufficiently accurate for nearly 
all practical purposes. 

35. Calculation of Exciting Current. Before drawing 
the complete transformer vector diagram, including the 
reactive drop calculated by means of the formula devel- 
oped in the preceding article, it is necessary to consider 



how the magnitude and phase of the exciting current 
component of the total primary current may be pre- 

The exciting current (TV) may be thought of as con- 
sisting of two components: (i) the magnetizing com- 
ponent (/o) in phase with the main component of the 
magnetic flux, i.e., that which links with both primary 
and secondary coils, and (2) the " energy " component 

Max 4 value of current component 

Amp, turns to produoe B max. 

Phaee of induced e. m.f. 

Total iron loss (watts) 

Primary impressed volts. 

FIG. 43. Vector Diagram showing Components of Exciting Current. 

(I w ) leading /o by one-quarter period, and, therefore, 
exactly opposite in phase to the induced e.m.f. 

The magnitude of this component depends upon the 
amount of the iron losses only, because the very small 
copper losses (I 2 e Ri) may be neglected. 

If these components could be considered sine waves, 
the vector construction of Fig. 43 would give correctly 
the magnitude and phase of the total exciting current L. 
For values of flux density above the " knee " of the 
B- H curve, the instantaneous values of the magnetizing 
current are no longer proportional to the flux, and this 


component of the total exciting current cannot therefore 
be regarded as a sine wave even if the flux variations 
are sinusoidal. The error introduced by using the con- 
struction of Fig. 43 is, however, usually negligible because 
the exciting current is a very small fraction of the total 
primary current. 

The notes on Fig. 43 are self explanatory, but reference 
should be made to Fig. 44 from which the ampere turns 
per inch of the iron core may be read for any value of 
the (maximum) flux density. The flux density is given 
in gausses, or maxwells per square centimeter of cross- 
section.* The total magnetizing ampere turns are equal 
to the number read off the curve multiplied by the mean 
length of path of the flux which links with both primary 
and secondary coils. When butt joints are present in 
the core, the added reluctance should be allowed for. 
Each butt joint may be considered as an air gap 0.005 m - 
long, and the ampere turns to be allowed in addition to 
those for the iron portion of the magnetic circuit are 


A mp. turns for joints 


= 9.01 X m ax XNo. of butt joints in series. (41) 

Instead of calculating the exciting current by the 
method outlined above, designers sometimes make use 

* The writer makes no apology for using both the inch and the centi- 
meter as units of length. So long as engineers insist that the inch has 
certain inherent virtues which the centimeter does not possess, they 
should submit without protest to the inconvenience ancl possible dis- 
advantage of having to use conversion factors, especially in connection 
with work based on the fundamental laws of physics. 















20 30 40 50 

Ampere-turns per inch 



FIG. 44. Curve giving Connection between Magnetizing Ampere-turns 
and Flux Density in Transformer Iron. 


of curves connecting maximum core density and volt- 
amperes of total exciting current per cubic inch or per 
pound of core; the data being obtained from tests on 
completed transformers. The fact that the total volt- 
amperes of excitation (neglecting air gaps) are some 
function of the flux density multiplied by the weight of 
the iron in the transformer core, may be explained as 
follows : 

Let w = total watts lost per pound of iron, correspond- 

ing to a particular value of B as read off 

one of the curves of Fig. 27; 
a = Ampere turns per inch as read off Fig. 44 ; 
A = cross-section of iron in the core, measured 

perpendicularly to the magnetic flux lines 

(square inches) ; 
/= Length of the core in the direction of the 

flux lines (inches) ; 
P = Weight of core in pounds = o.2&Al. 

The symbols previously used are: 
T p = number of primary turns: 

Given definite values for B and /, the " in phase " 
component of the exciting current is 

T core loss wXP 


and the " wattless " component, or true magnetizing 
current, is 




Multiplying both sides of the equation by ~, we get 

Epl e _ volt-amperes of total excitation 
P weight of core 


This formula may be used for plotting curves such 
as those in Fig. 45. Thus, if 

= 13,000 gausses, 

/=6o cycles per second, 
20 = 1.55 (read off curve for silicon steel in Fig. 27), 

a = 22 (from Fig. 44); and, by Formula (42) 

Volt-amperes per pound 

The error in this method of deriving the curves of 
Fig. 45 is due to the fact that sine waves are assumed. 
The data for plotting the curves should properly be 
obtained from tests on cores made out of the material 
to be used in the construction of the transformer. 







* 11000 




I 9000 





5 10 15 20 25 30 35 40 

Exciting volt-aiperes per Ib. of stampings 
(Approximate values for either iron or silicon-steel) 

FIG. 45. Curves giving Connection between Exciting Volt-amperes 
and Flux Density in Transformer Stampings. 


The effect of the magnetizing current component in 
distorting the current waves may be appreciable when 
the core density is carried up to high values. The curve 
of flux variation cannot then be a sine wave, and the 
introduction of high harmonics in the current wave may 
aggravate the disturbances that are always liable to 
occur in telephone circuits paralleling overhead trans- 
mission lines. This is one reason why high values of 
the exciting current are objectionable. An open-circuit 
primary current exceeding 10 per cent of the full-load 
current would rarely be permissible. 

36. Vector Diagrams Showing Effect of Magnetic 
Leakage on Voltage Regulation of Transformers. The 
t vector diagrams, Figs. 46, 47, and 48, have been drawn 
to show the voltage relations in transformers having 
appreciable magnetic leakage. The proportionate length 
of the vectors representing IR drop. IX drop, and 
magnetizing current, has purposely been exaggerated in 
order that the construction of the diagrams may be 
easily followed. 

Fig. 46 is the complete vector diagram of a transformer ; 
the meaning of the various component quantities being 
as follows: 

2 = Induced secondary e.m.f., due to the flux (OB) 
linking with the secondary turns; 

E s = Secondary terminal voltage when the secondary 
current is I s amperes on a load power factor 
of cos 6] 

I e = Primary exciting current, calculated as ex- 
plained in the preceding article; 

/i = Balancing component of total primary current 

IP = Total primary current ; 
E'\ = Balancing component of induced primary volt- 

PE'\ = IR drop due to primary resistance (drawn 

parallel to 01 P ) ; 
EpP = IX drop due to leakage reactance (drawn at 

right angles to OI P ) ; 
E v Impressed primary e.m.f . 

FIG. 46. Vector Diagram of Transformer on Inductive Load. 

It is usually permissible to neglect the exciting current 
component when considering full-load conditions. This 
leads to the simpler diagram, Fig. 47, in which the total 
primary current is supposed to be of the same magnitude 
and phase as what has previously been referred to as the 
balancing component of the total primary current. 

The dotted lines in Fig. 47 show how a still greater 
simplification may be effected in drawing a vector 



diagram from which the voltage regulation can be cal- 
culated. Instead of drawing the two vectors QEi 
and OE S for the induced and terminal secondary voltages, 
we can draw OE ( opposite in phase to and equal to 

Then E e P (drawn parallel to 01 1) is the 

component of the impressed primary volts necessary 
to overcome the ohmic resistance of both primary and 
secondary windings. 


Secondary Risistance >- 

FIG. 47. Simplified Vector Diagram of Transformer; Exciting Current 

It is now only necessary to turn this diagram through 
1 80 degrees, and eliminate all unnecessary vectors, in 
order to arrive at the very simple diagram of Fig. 48, 
from which the voltage regulation can be calculated. 

37. Formulas for Voltage Regulation. From an 
inspection of Fig. 48, it is seen that 

(IiR p )+EeCasO 

COS <b 

. . (43) 



wherein cos is known (being the power factor of the 
external load), and cos <f> has not yet been determined. 


( . 
* ' ' (44) 

FIG. 48. Simple Transformer Vector Diagram for Calculation of 
Voltage Regulation. 

which can be used to calculate and therefore cos>. 
The percentage regulation is 

, (45) 

or, if the ohmic drop is expressed as a percentage of the 
(lower) terminal voltage: 
Per cent regulation 

_Per cent equiv. IR drop +100 (cos 9 cos #) / ,\ 



The difference between the angles 6 and (Fig. 
48) is generally small, and it is then permissible to 
assume that OD = OE P . But 

E e +IiR cos d+IiX p sin 6, 

Per cent regulation (approximate) 

= Per cent IR cos 0+per cent IX sin 0. (47) 

If the power factor were leading instead of lagging as 
in Fig. 48, the plus sign would have to be changed to a 
minus sign. 

Example. In order to show that the approximate 
Formula (47) is sufficiently accurate for practical pur- 
poses, the following numerical values are assumed. 

Power factor (cos 0) =0.8. 
Total IR drop = 1.5 per cent. 
Total IX drop = 6.0 per cent. 

By Formula (44), 


tan<= =0.81, 

whence cos <f> = 0.777, an d> by Formula (46), 

Regulation = '-U- = 4.9 per cent. 


By the approximate Formula (47), 

Regulation = (i.sXo.8) + (6Xo.6)=4.8 per cent. 


The total equivalent voltage drop, due to the resistance 
of the windings (the quantity I\R P of the vector dia- 
grams) is usually between i and 2 per cent of the ter- 
minal voltage in modern transformers. The reactive 
voltage drop caused by magnetic leakage (the quantit}^ 
IiXp in the vector diagrams) is nearly always greater 
than the IR drop, being 3 to 8 per cent of the terminal 
voltage. Sometimes it is TO per cent, or even more, 
especially in high- voltage transformers where the space 
occupied by insulation is considerable, or in transformers 
of very large size, when the object is to keep the current 
on short circuit within safe limits. 



37. The Output Equation. The volt-ampere output 
of a single-phase transformer is E X / which, as explained 
in Art. 6, may be written 

Volt-amperes = x$X(TI), . . (48) 

where TI stands for the total ampere turns of either the 
primary or secondary winding. 

There is no limit to the number of designs which will 
satisfy this equation; the total flux, 3>, is roughly a 
measure of the cross-section of the iron core, while the 
quantity (TI) determines the cross-section of the wind- 
ings. The problem before the designer is to proportion 
the parts and dispose the material in such a way as to 
obtain the desired output and specified efficiency at the 
lowest cost. The temperature rise is also a matter of 
importance which must be watched, and light weight 
is occasionally more important than cost. 

It cannot be said that there is one method of attacking 
the problems of transformer design which has indisputable 
advantages over all others; and in this, as in all design, 
the judgment and experience of the individual designer 
must necessarily play an important part. The apparent 



simplicity of the calculations involved in transformer 
design is the probable cause of the many more or less 
unsuccessful attempts to reach the desired end by purely 
mathematical methods. It is not possible to include 
all the variable factors in practical mathematical equa- 
tions purporting to give the ideal quantities and pro- 
portions to satisfy the specification. Methods of pro- 
cedure aiming to dispense with individual judgment and 
a certain amount of correction or adjustment in the final 
design, should generally be discountenanced, because 
they are based on inade'quate or incorrect assumptions 
which are liable to be overlooked as the work proceeds 
and becomes finally crystallized into more or less for- 
midable equations and formulas of unwieldy propor- 

No claim to originality is made in connection with 
the following method of procedure; indeed it is ques- 
tionable whether the mass of existing literature treating 
of the alternating current transformer leaves anything 
new to be said on the subject of procedure in design. 
All that the present writer hopes to present is a treat- 
ment consistent with what has gone before, based always 
on the fundamental principles of physics even though 
the use of empirical constants may be necessary. 

Instead of attempting to take account at one time of 
all the conditions to be satisfied in the final design, the 
factors which have the greatest influence on the dimen- 
sions will be considered first; items such as temperature 
rise and voltage regulation being checked later and, if 
necessary, corrected by slight changes in the dimensions 
or proportions of the preliminary design. 


38. Specifications. It will be advisable to list here 
the particulars usually specified by the buyer, and sup-, 
plement these, if required, with certain assumptions that 
the manufacturer must make before he can proceed 
with a particular design. 

(1) K.v.a. output. 

(2) Number of phases. 

(3) Primary and secondary voltages (E p and E s ). 

(4) Frequency (/). 

(5) Efficiency under specified* conditions. 

(6) Voltage regulation under specified load. 

(7) Method of cooling Temperature rise. 

(8) Maximum permissible open-circuit exciting cur- 

Items (i) to (4) must always be stated by the pur- 
chaser, while the other items may be determined by the 
manufacturer, who should, however, be called upon to 
furnish these particulars in connection with any competi- 
tive offer. 

With reference to item (5), if the efficiency is stated 
for two different loads, the permissible copper and iron 
losses can be calculated. If the buyer does not furnish 
these particulars, he should state whether the trans- 
former is for use in power stations or on distributing 
lines, in order that the relation of the iron losses to the 
total losses may be adjusted to give a reasonable all-day 
efficiency. In any case, before proceeding with the 
design, the maximum permissible iron and copper losses 
must be known or assumed. 

The requirements of items (6), (7). and (8), are to some 


extent satisfied, even in the preliminary design, by 
selecting a flux density (B) and a current density (A) 
from the values given in Article 20, because industrial 
competition and experience have shown these values 
to give the best results while using the smallest per- 
missible amount of material. Thus, by selecting a 
proper value for A, both the local heating and the IR 
drop of the windings will probably be within reason- 
able limits. The other factor influencing the voltage 
regulation (item (6)) is the reactive drop, which can 
generally be controlled by suitably subdividing the 

A proper value of the flux density (B) will generally 
keep item (8) within the customary limits. 

39. Estimate of Number. of Turns in Windings. Re- 
turning to the Formula (48) in Article 37, if a suitable 
value for T could be determined or assumed, the only 
unknown quantity in the output equation would be 3> 
and we should then have a starting-point from which the 
dimensions of a preliminary design could be easily cal- 

Let T^_= volts per turn (of either primary or sec- 
ondary winding) then, in order to express this quan- 
tity in terms of the volt-ampere output, we have, 


from which T must be eliminated, since the reason 
for seeking a value for V t is that T may be calculated 


Using the value of (El) as given by Formula (48), 
we can write 

TI TYXio 8 ' 

V t = Vvolt-ampere output X - (49) 

The quantity in brackets under the second radical is 
found to have an approximately constant value, for an 
efficient and economical design of a given type, without 
reference to the output. This permits of the formula 
being put in the form 

V t = cX Vvolt-ampere output, .... (490) 

where c is an empirical coefficient based on data taken 
from practical designs. 

Factors Influencing the Value of the Coefficient c. 

\ f ^ 
It is proposed to examine the meaning of the ratio 

which appears under the second radical of Formula (49) 
with a view to expressing this in terms of known quan- 
tities, or of quantities that can easily be estimated. 
Let W c = full load copper losses (watts) ; 

Wi = core losses (watts) ; 
the relation between these losses being; 



wherein b must always be known before proceeding 
with the design. 

Let l c = mean length per turn of copper in windings ; 
/i = mean length of magnetic circuit measured 

along flux lines; 

W c = constant X A 2 X volume of copper i 

where k c is a constant to be determined later. Similarly 
Wi = constant XfB n Xvolume of iron 


wherein k L is another constant to be determined later. 

Inserting these values in Formula (50), the required 
ratio can be put in the form 

( } 
TI bk,B n - l \l 

This ratio is thus seen to depend on certain quantities 
and constants which are only slightly influenced by the 
output of the transformer. They -depend on such items 
as the ratio of copper losses to iron losses (i.e., whether 
the transformer is for use on power transmission lines, or 
distributing circuits); temperature rise and methods of 


cooling; space factor (voltage); and also on the type 
whether core or shell since this affects the best relation 
between mean lengths of the copper and iron circuits. 

The Factor k c . Using the inch for the unit of length, 
and allowing 7 per cent for eddy-current losses in the 
copper, the resistivity of the windings will be 0.9X10" 
ohms per inch-cube at a temperature of 80 C.; the 
loss per cubic inch of copper = A 2 X 0.9X10-, and 

since the volume is 2( j/ c , it follows that k c = 2X 

o.9Xio~ 6 . 

The Factor ki If w = total watts lost per pound as 
read off one of the curves of Fig. 27, and if h is in inches, 
we have the equation 




The Factor b. The ratio of full-load copper loss to 
iron loss will determine the load at which maximum 
efficiency occurs. 

Let us assume the k.v.a. output and the frequency of 
a given transformer to be constant, and determine the 
conditions under which the total losses will be a minimum. 
It is understood that, if the current / is increased, the 
voltage. E, must be decreased; but the condition k.v.a, 
= El must always be satisfied. 


The sum of the losses is PFc+H^; but 

(k.v.a.) 2 a constant 

2 2 


Also, since/ remains constant, EccB, and we can write 

Wi = a constant X# n . 
The quantity which must be a minimum is therefore 

a constant . . 

ha constantX-E . 

If we take the differential coefficient of this function 
of E and put it equal to zero, we get the relation 

W~ 2 

The value of n for high densities is about 2, while 
for low densities it is nearer to 1.7, a good average 
being 1.85. Thus, to obtain maximum efficiency at 
full load in a power transformer, the ratio of copper 

loss to iron loss should be about b = -^-- =0.925. 

In a distributing transformer, in order to obtain a 
good all-day efficiency, the maximum efficiency should 
occur at about f full load, whence 

W t 


Taking 71 = 1.75, because of the lower densities generally 
used in small self-cooling transformers, we get 

, 1.75X9 / x 

b= -=61.97 or (say) 2. 

The Ratio -. Considerable variations in this ratio are 


permissible, even in transformers of a given type wound 
for a particular voltage, and that is one reason why a 
close estimate of the volts per turn as given by Formula 
(49) is* not necessary. Refinements in proportioning the 
dimensions of a transformer are rarely justified by any 
appreciable improvement in cost or efficiency; a certain 
minimum quantity of material is required in order to 
keep the losses within the specified limits; but consid- 
erable changes in the shape of the magnetic and electric 
circuits can be made without greatly altering the total 
cost of iron and copper, provided always that the im- 
portant items of temperature rise and regulation are 
checked and maintained within the specified limits. 

Figs. 49 and 50 show the assembled iron stampings of 
single-phase shell- and core- type transformers. The 
proportions will depend somewhat upon the voltage and 
method of cooling; but if the leading dimensions are 
expressed in terms of the width (L) of the stampings 
under the coils, they will generally be within the following 


Shell T3>pe. Core Type. 

S 2 to 3 times L S = i to i . 8 times L 

B = o.$ to o . 75 times L B = i to i . 5 times L 

D = o.6 to 1.2 times L D = i to 2 times L 

H = i . 2 to 3 5 times L H = 3 to 6 times L 



By taking the averages of these figures, and roughly 
approximating the lengths l c and k in each case, the mean 
value cf the required ratio is found to be 

" 7 = 1.2 (approx.) for shell type, ] 

7 = 0.3 (approx.) for core type. 

FIG. 4Q. Assembled Stampings of Single-phase Shell-type Transformer. 

Having determined the values of the various quan- 
tities appearing in Formula (53), it is now possible 
to calculate an approximate average value for the 


quantity ^ and for the coefficient c of Formula (49). 

We shall make the further assumptions (refer Art. 20) 
that A = 1100 amperes per square inch, and ^=8000 



gausses; the transformer being of the shell type for use 
on distributing circuits of frequency 60. Then, by 
Formula (53), ^ g 

/$ _^<o79X i IPO X i. 2X6.45X60X9000 _ 

~Ti~ ~io 6 x 2x0.28x0.75 

= 19,720 

FIG. 50. Assembled Stampings of- Single-phase Core-type Transformer. 

wherein the figure 0.75 is the value of w read off the 
curve for silicon steel in Fig. 27. 

The value of the coefficient in Formula (49), for the 
assumed conditions, is therefore 


X 19, 720 = 0.0296 


Similarly, for a core-type power transformer; if 7=25. 
B = 13,000, and A = 1350, we have, 

/$ = 2X0.9X1350X0.5X6.45X25X13,000^ 6 
TI io 6 Xo.925Xo.28Xo.58 

Whence c = 0.02 74. 

Having shown what factors determine this design 
coefficient, it will merely be necessary to give a list of 
values from which a selection should be made for the 
purpose of calculating the quantity V t of Formula (490).- 

For shell-type power transformers = 0.04 to 0.045 

For shell- type distributing transformers ^ = 0.03 
For core- type power transformers = 0.025 to 0.03 

For core- type distributing transformers c = o.o2 

Where a choice of two values of c is given, the lower 
value should be chosen for transformers wound for high 
pressures. When the voltage is low the value of c is 
slightly higher because of the alteration in the ratio. 

- which depends somewhat on the copper space factor. 

The proposed values here given for this design coeffi- 
cient are based on the assumption that silicon steel 
stampings are used in the core. If ordinary trans- 
former iron is used as, for instance, in small distrib- 
uting transformers it will be advisable to take about J 
of the above values for the coefficient c. 

40. Procedure to Determine Dimensions of a New 
Design. With the aid of the design coefficient c, it is 
now possible to calculate the number of volts that should 


be generated in one turn jf the winding of a transformer 
of good design according to present knowledge and prac- 
tice. The logical sequence of the succeeding steps in 
the design, may be outlined as follows: 

(1) Determine approximate dimensions. 

(a) Calculate volts per turn by Formula (49). 

(b) Assume current density (select suitable trial 

value from table in Art. 20). Decide on 
number of coils. Calculate cross-section 
of copper. 

(c) Decide upon necessary insulation and oil- 

or air-ducts between coils, and between 
windings and core. Determine shape and 
size of " window " or opening necessary to 
accommodate the windings. 

(d) Calculate total flux required. Assume flux 

density (select suitable trial value from 
table in Art. 20), and calculate cross-sec- 
tion of core. Decide upon shape and size 
of section, including oil- or air-ducts if 

(e) Calculate iron and copper losses, and modify 

the design slightly if necessary to keep 
these within the specified limits. 

(2) Calculate approximate weight and cost of iron 
and copper if desired to check with permissible maximum 
before proceeding with the design. 

(3) Calculate exciting current. 

(4) Calculate leakage reactance and voltage regula- 


. (5) Calculate necessary cooling surfaces. Design con- 
taining tank and lid, providing not only sufficient oil 
capacity and cooling surface, but also the necessary 
clearances to insure proper insulation between current- 
carrying parts and the case. Calculate temperature rise. 

41. Space Factors. The copper space factor, as pre- 
viously defined (see Art. 15), is the ratio between the 
cross-section of copper and the area of the opening or 
" window " which is necessary to accommodate this 
copper together with the insulation and oil- or air-ducts. 
It may vary" between 0.55 in transformers for use on 
circuits not exceeding 660 volts, to 0.06 in power trans- 
formers wound for about 100,000 volts. An estimated 
value of the probable copper space factor may be useful 
to the designer when deciding upon one of the dimensions 
of the " window " in the iron core. For this purpose, 
the curves of Fig. 51 may be used, although the best 
design and arrangement of coils and ducts will not always 
lead to a space factor falling within the limits included 
between these two curves. 

Iron Space Factor. The so-called stacking factor for 
the iron core will be between 0.86 and 0.9, and the total 
thickness of core, multiplied by this factor, will give 
the net thickness of iron if there are no oil- or air-ducts. 
When spaces are left between sections of the core for 
air or oil circulation, the iron space factor may be from 
0.65 to 0.75. 

42. Weight and Cost of Transformers. The weight 
per k.v.a. of transformer output depends not only upon 
the total output, but also upon the voltage and fre- 
quency. The net and gross weights of particular trans- 



S| \ 

I : 


formers can be obtained from manufacturers' catalogues 
and also from the Handbooks for Electrical Engineers. 
The effect of output and frequency on the weight of a 
line of transformers designed for a particular voltage 
(in this instance, 22,000 volts) is roughly indicated by 
the following figures of weight per k.v.a. of output. 
These figures include the weight of oil and case. 

{ 100 k.v.a. output 40 Ib. 

Frequency 60 \ 

[ 500 k.v.a. output 23 Ib. 

^ f 100 k.v.a. output 152 Ib. 

Frequency 25 \ 

[ 500 k.v.a. output 35 Ib. 

The cost of transformers, depending as it does on the 
fluctuating prices of copper and iron, is very unstable. 
Within the last few years, the variation in the price of 
copper wire has been about 100 per cent, and the cost of 
the laminated iron for the cores has also undergone great 
changes. The best that can be done here is to indicate 
how the cost depends upon voltage and output. That a 
high frequency always means a cheaper transformer is 
evident from an inspection of the fundamental Formula 
(48) of Art. 37. If/ is increased, either 3>, or (TI), or 
both, can be reduced, and this means a saving of iron, 
or copper, or both. The effect of an increase in voltage 
is felt particularly in the smaller sizes, but an increase 
of voltage always means an addition to the cost; while 
an increase of size for a given voltage results in a reduc- 
tion of the cost per k.v.a. of output. 

Some idea of the dependence of cost on output and 
voltage may be gained from the fact that the unit cost 


would be about the same for (i) a 1500 k.v.a. trans- 
former wound for 22,000 volts, (2) a 2000 k.v.a. trans- 
former wound for 44,000 volts, and (3) a 3000 k.v.a. 
transformer wound for 88,000 volts. 

Three-phase Transformers. It does not appear to be 
necessary to supplement what has been said in Articles 
5 and 8 on the subject of three-phase transformers. 
Once the principles underlying the design of single-phase 
transformers are thoroughly understood, it is merely 
necessary to divide any polyphase transformer (see 
Figs. 12, 13, and 14) into sections which can be treated 
as single-phase transformers, due attention being paid 
to the voltage and k.v.a. capacity of each such unit 
section of the three-phase transformer. 

The saving of materials effected by combining the 
magnetic circuits of three single-phase transformers 
so as to produce one three-phase unit, usually results 
in a reduction of 10 per cent in the weight and 

43. Numerical Example. It is proposed to design a 
single-phase 1500 k.v.a. oil-insulated, water-cooled, 
transformer for use on an 88,000- volt power transmission 
system. A design sheet containing more detailed items 
than would generally be considered necessary will be 
used in order to illustrate the various steps in the design 
as developed and discussed in the preceding articles. 
Two columns will be provided for recording the known or 
calculated quantities, the first being used for preliminary 
assumptions or tentative values, while the second will be 
used for final results after the preliminary values have 
been either confirmed or modified. 



Output ............ ' .................. i ,500 k.v.a. 

Number of phases ................. .... one 

H.T. voltage ......................... 88,000 

L.T. voltage .......................... 6,000 

Frequency ........................... 50 

Maximum efficiency, to occur at full load 

and not to be less than .............. 98 . i% 

Voltage regulation, on 80 per cent power 

factor ............................. 5% 

Temperature rise after continuous full- 

load run ......................... . . 40 C. 

Test voltage: H.T. winding to case and 

L.T. coils .......................... 177,000 

L.T. winding to case .................. 14,000 

The calculated values of the various items are here 
brought together for reference and for convenience in 
following the successive steps in the design. The items 
are numbered to facilitate reference to the notes and more 
detailed calculations which follow. 

Items (i) and (2). L.T. Winding. By Formula (490), 
Art. 39, page 142, the volts per turn, for a shell-type 
power transformer, are 

500,000 = 51.5, 

r s = 66oo = I28 . 




i. Volts per turn. 


Total number of turns 

Number of coils 

Number of turns per coil . 

Secondary current, amperes 

6. Current density, amperes per sq. in. . . 

7. Cross-section of each conductor, sq. in. 

8. Insulation on wire, cotton tape, in. . . 

9. Insulation between layers, in 

10. Number of turns per layer, per coil. . . 

11. Number of layers 

12. Overall width of finished coil (say), in. 

13. Thickness (or depth) of coil, with 

allowance for irregularities and 
bulging at center, in 


14. Total number of turns 

15. Number of coils 

16. Number of turns per coil 

17. Primary current, amperes 

18. Current density, amperes per sq. in. . 

19. Cross-section of each wire, sq. in. . . . 

20. Insul. on wire (cotton covering), in. . . 

21. Insul. between layers, fullerboard, in. . 

22. Number of turns per layer, per coil. . . 

23. Number of layers; in all but end coils 

24. Overall width of finished coil, in 

25. Thickness or depth of coil, in 

26. Make sketch of assembly of coils, with 

necessary insulating spaces and oil 


or Approxi- 










3strips, eacho. 16X0.3=0.144 

| 0.026 

2X0. 006) +o .012 = 0. 024 



. 5 


80 in 2 coils; 95 in 16 coils 
1 7 . 05 

0.04X0. 26 = 0.0104 
2X0.008 = 0.016 









or Approxi- 


27. Size of " window " or opening for 
windings, in 

12. 75X32 

28 Total flux (maxwells) 


2.36Xio 7 

29. Maximum value of flux density in 
core under windings (gausses) 
30 Cross-section of iron under coils, sq. in. 




31 Number of oil ducts in core 


32. Width of oil ducts in core 
33. Width of stampings under windings, 




34 Net length of iron in core in 


35. Gross length of core, in 
36. Cross-section of iron in magnetic cir- 
cuit outside windings sq in. . . 



37. Flux density in core outside windings 



38. Average length of magnetic circuit 
under coils, in 

in. s 

39. Average length of magnetic circuit 
outside coils in 

III 1 

40 Weight of core, Ib 



41 Losses in the iron, watts 



42 Mean length per turn of primary, ft. . 

IO 11 

43. Resistance of primary. winding, ohms . 
44. Full-load losses in primary (exc. cur- 
rent neglected) . . 




45. Mean length per turn of secondary, ft. 

IO. It 

46. Resistance of secondary winding, ohms 
47 Full-load losses in secondary watts 



48. Total full-load copper losses, watts . . 

W c 

10,220 . 





or Approxi- 


49. Total weight of copper in windings, Ib. 
50. Efficiency at full load ' (unity power 

o 08 5 

51. Efficiency at other loads and power 
factors (refer to text following). 
52. No-load primary exciting current, 
amperes . . 


2 I ^ 

53. Reactive voltage drop 

IiX v 


54. Equivalent ohmic voltage drop 



55. Regulation on unity power factor, per 
cent. . 

O 7^t? 

56. Regulation on 80 per cent power fac- 
tor, per cent. 

2 ^ 

57. Effective cooling surface of tank, sq. in. 
58. Number of watts dissipated from tank 



59. Watts to be carried away by circula- 
ting water 


60. Size and length of pipe in cooling coil,. 
61. Approximate flow of water per minute, 


4. 37 

62. Approximate weight of oil Ib 

7 3OO 

63. Estimated total weight of transformer, 
Ib. . 


(3) and (4). The number of separate coils is 
determined by the following considerations: 

(a) The voltage per coil should preferably not exceed 
5000 volts. 


(b) The thickness per coil should be small (usually 
within 1.5 in.) in order that the heat may readily be 
carried away by the oil or air in the ducts between coils 
(Refer Art. 23). 

(c) The number of coils must be large enough to admit 
of proper subdivision into sections of adjacent primary 
and secondary coils to satisfy the requirements of regu- 
lation by limiting the magnetic flux-linkages of the leak- 
age field. 

(d) An even number of L.T. coils is desirable in order 
to provide for a low- tension coil near the iron at each end 
of the stack. 

To satisfy (a), ^here'must be at least - - or, say- 

18 H.T. coils. If an equal number of secondary coils 
were provided, we could, if desired, have as many as 
eighteen similar high-low sections which would be more 
than necessary to satisfy (c). The number of these 
high-low sections or groupings must be estimated now 
in order that the arrangement of the coils, and the num- 
ber of secondary coils, may be decided upon with a view 
to calculating the size of the " windows " in the mag- 
netic circuit. It is true that the calculations of reactive 
drop and regulation can only be made later; but these 
will check the correctness of the assumptions now made, 
and the coil grouping will have to be changed if neces- 
sary after the preliminary design has been carried some- 
what farther. The least space occupied by the insula- 
tion, and the shortest magnetic circuit, would be obtained 
by grouping all the primary coils in the center, with half 
the secondary winding at each end, thus giving only 


two high-low sections ; but this would lead to a very high 
leakage reactance, and regulation much worse than the 
specified 6 per cent. Experience suggests that about six 
high-low sections should suffice in a transformer of this 
size and voltage, and we shall try this by arranging the 
high-tension coils in groups of six, and providing six 
secondary coils (see Fig. 52). This gives us for item 
(4), J -f- = 2i.3 or, say, 21, whence T s = i26. 
Items (5) to (13). The secondary current is 7 S = 

I? ^ 00 ' 000 = 227 amperes. From Art. 20, we select 

A = 1600 as a reasonable value for the current density, 
.giving --=0.142 sq. in. for the cross-section of the 


secondary conductor. 

In order to decide upon a suitable width of copper 
in the secondary coils, it will be desirable to eti- 
mate the total space required for the windings so 
that the proportions of the " window " may be such 
as have been found satisfactory in practice. The 
space factor (Art. 41) is not likely to be better than 
o.i, which gives for the area of the "window" 

2Xi26~Xo.i42 u . . , ., 

= 358 sq. in. Also, if a reasonable as- 

sumption is that H = 2.5 times D (see Fig. 49, page 147), 
it follows that 2.5Z>X.D = 358; whence Z) = i2 inches. 

The clearance between copper and iron under oil, 
for a working pressure of 6600 volts (Formula (12), 
Art. 16), should be about 0.25+0.05X6.6 = 0.58 in. 
For the insulation between layers, we might have 
0.02 in. for cotton, and a strip of 0.012 in. fullerboard, 


making a total of 21X0.032=0.67 in. The thickness 
of each secondary conductor will therefore be about 

12 -(0.58+0.67+0.58) . , . , . 

-=0.485 in., which gives a width 

of ai ^ 2 =0.293 in. Let us make this 0.3- in., and 

build up each conductor of three strips 0.16 in. thick, 
with 0.006 paper between wires (to reduce eddy cur- 
rent loss) and cotton tape outside. Allowing 0.026 in. 
for the cotton tape, and 0.012 in. for a strip of fuller- 
board between turns, the total thickness of insulation, 
measured across the layers, is 21 X (0.026 +0.024) = 
1.05 in. 

A width of " window " of 12.75 m - ( see Fig. 52) 
will accommodate these coils. The current density- 
with this size of copper is 

Items (14) to (25). H.T. Winding. T p 


= 1680. This may be divided into 16 coils of 95 turns 
each, and 2 coils of only 80 turns each, which would 
be placed at the ends of the winding and provided with 
extra insulation between the end turns (see Art. 14). 

According to Formula (13) of Ait. 16, the thickness 
of insulation consisting of partitions of fullerbcard 
with spaces between for oil circulation separating the 
H.T. copper from L.T. coils or grounded ircn, should 
not be less than 0.25+0.03X88 = 2 89 in. Let us make 
this clearance 3 in. Then, since the width of opening 



is 12.75 m " the maximum permissible depth of winding 
of the primary coils will be 12.75 6 = 6.75 in. The 

N . T 1,500,000 

primary current (Item 17) is /*=-^- J = 17-05 amps. 


(approx.). The cross-section of each wire is 

= 0.01065 sq. in. Allowing 0.016 in. for the total increase 
of thickness due to the cotton insulation, and 0.012 

FIG. 52. Section through Windings and Insulation. 

in. for a strip of fullerboard between turns, the thick- 
ness of the copper strip (assuming flat strip to be used) 

must not exceed ( ) 0.028 = 0.043 in., which 
\ 95 / 

makes the width of copper strip equal to - ^ =0.248 


in. Try copper strip 0.26X0.04=0.0104 sq. in., making 
A = 1640. 


The two end coils, with fewer turns, would be built 
up to about the same depth as the other coils by putting 
increasing thicknesses of insulation between the end 
turns. Thus, since there is a total thickness of copper 
equal to 0.04 X (95 80) =0.6 in. to be replaced by insula- 
tion, we might gradually increase the thickness of fuller- 
board between the last eight turns from 0.012 in. to 
0.15 in. 

Items (26) and (27). Size of Opening lor Windings. 
A drawing to a fairly large scale, showing the cross- 
section through the coils and insulation, should now be 
made. Oil ducts not less than \ in. or ^ in. wide 
should be provided near the coils to carry off the heat, 
and the large oil spaces between the H.T. coils and the 
L.T. coils and iron stampings, should be broken up by 
partitions of pressboard or other similar insulating 
material, as indicated roughly in a portion of the sketch, 
Fig. 52. In this manner the second dimension of the 
" window " is obtained. This is found to be 32 in., 
whence the copper space factor is 

(1680X0.0104) + (126 0<o_i445) _ 

Items (28) to (41). The Magnetic Circuit. By 
Formula (i), Art. 2, 

88,000 X io 8 

Before assuming a flux density for the core, let us 
calculate the permissible losses. 


The full load efficiency being 0.981, the total losses 

i, 500,000 X(i 0.081) 

are -^ ^- = 29,000 watts. Also, since 



the ratio is approximately 0.925 (see Art. 39, 

under sub-heading The Factor b) y it follows that 

W c = -i = 1 5 , TOO watts. 


W c = 29,000 15,100 = 13,900 watts. 

Let us assume the width of core under the windings 
(the dimension L of Fig. 49) to be 1 1 in. and the width, 
B, of the return circuit carrying half the flux, to be 
5.5 in. Then the average lengtfi of the magnetic circuit, 
measured along the flux lines, will be 2(12.75 + 5.5+32 + 

5-5) = 111-5 in - 

If the flux density is taken at 13,000 gausses (selected 
from the approximate values of Art. 20) the cross- 


section of the iron is ^ = 282 sq. in. The 

watts lost per pound (from Fig. 27) are w = 1.27, whence 
the total iron loss is 

, */' 'h ^ *~* 

1^ = 1.27X0.28X282X111.5 = 11,200 watts, 

which is considerably less than the permissible loss. It is 
not advisable to use flux densities much in excess of the 
selected value of 13,000 gausses for the following reasons: 
(a) The distortion of wave shapes when the mag- 
netization is carried beyond the " knee " of the B-H 


(b) The large value of the exciting current. 

(c) The difficulty of getting rid of the heat from the 
surface of the iron when the watts lost per unit volume 
are considerable. 

Let us, therefore, proceed with the design on the basis 
of 14,000 gausses as an upper limit for the flux density. 

If no oil ducts are provided between sections of the 
stampings, the stacking factor will be about 0.89. A 
gross length of 27 in. (Item 35) gives 24 in. for the net 
length, and a cioss-section of 24 X 1 1 = 264 sq. in. Whence 
$ = 13,850 gausses, and the total weight of iron is 
264X111. 5X0.28 = 8250 Ib. 

The watts per pound, from Fig. 27, are ^=1.44, 
whence Wi = 1 1 ,900. 

Items (42) to (49), Copper Loss. The mean length per 
turn of the windings is best obtained by making a draw- 
ing such as Fig. 53. This sketch shows a section through 
the stampings parallel with the plane of the coils. The 
mean length per turn of the secondary, as measured 
off the drawing, is 122 in., and since the length per turn 
of the primary coils will be about the same, this dimen- 
sion will be used in both cases. Taking the resistivity 
of the copper at 0.9X10" ohms per inch cube (see The 
Factor k c , in Art. 39), the primary resistance (hot) is 

D 0.9X122X1680 
/Ci = - - = 18.1 ohms, 

i o 6 X 0.0104 

whence the losses (Item 44) are 

(i7.o5) 2 X 18.1 = 5260 watts. 


For the secondary winding we have 

R2 = -=0.0062 ohm. 

io 6 X 0.144 



FIG. 53. Section through Coil and Stampings. 

whence the losses (Item 47) are (227)2 Xo.og62 =4960 
watts, and 

^ = 5360+4960=10,220 watts, 
which is appreciably less than the permissible copper loss. 


It is at this stage of the calculations that changes 
should be made, if desirable, to reduce the cost of mate- 
rials, by making such modifications as would bring the 
losses near to the permissible upper limit. The obvious 
thing to do in this case would consist in increasing the 
current density in the windings, and perhaps making a 
small reduction in the number of turns. A considerable 
saving of copper would thus be effected without neces- 
sarily involving any appreciable increase in the weight 
of the iron stampings. Since this example is being 
worked through merely for the purpose of illustrating 
the manner in which fundamental principles of design 
may be applied in practice, no changes will be made 
here to the dimensions and quantities already calculated. 

The weight of copper (Item 49) is 

0.32 (122X1680X0.0104) + 

0.32(122X126X0.144) = 1,700 Ib. 

Items (50) and (51). Efficiency. The full-load effi- 
ciency on unity power factor is 



The calculated efficiencies at other loads are: 

At ij full load 0.985 

At f full load 0.984 

At | full load 0.981 

At | full load 0.968 


The full-load efficiency on 80 per cent power factor is 


= 0.982. 

Item (52). Open-circuit Exciting Current. Using 
the curves of Fig. 45 (see Art. 35 for explanation), we 
obtain for a density = 13,850 the value 23 volt-amperes 
per pound of core. The weight of iron (Item 40) being 
8250 lb., it follows that the exciting current is 

r 8250X23 

This is 12.6 per cent of the load component, which is 
rather more than it should be. If the design is altered, 
as previously suggested, to reduce the amount of copper, 
this will result in a reduction of the opening in the iron, 
and, therefore, also of the length of the magnetic circuit. 
It is, however, clear that the flux density (Item 29) must 
not be higher than 13,850 gausses. If the design were 
modified, it might be advisable to reduce this value by 
slightly increasing the cross-section of the magnetic 
cirduit. The fact that the exciting current component 
is fairly large relatively to the load current will lead to a 
small increase in the calculated copper loss (Item 44); 
but for practical purposes it is unnecessary to make the 

Items (53) to (56) Regulation. Referring to Fig. 52, it 
is seen that there are six high-low sections, all about 
equal, since the smaller number of turns in two out of 
eighteen primary coils is not worth considering in calcu- 


lations which cannot in any case be expected to yield 
very accurate results. The quantities for use in Formula 
(40) of Art. 34 have, therefore, the following values: 

Ti= :L * s -=2*o', 
71 = 17.05; 

7 = 10.15X12X2.54 = 310 cm.; 

# = 3X2.54 = 7.62 cm.; 

# = 1.7X2.54 = 4.32 cm.; 

5 = 0.38X2.54 = 0.965 cm.; 
h= 12.75X2.54 = 32.4 cm. 

whence the induced volts per section are, 
7iXi=475 volts. 

Since there are six sections, and all the turns are in series, 
the total reactive drop at full load is 

IiXp = 475 X 6 = 2850 volts, 

which is only 3 . 24 per cent of the primary impressed 

By Formula (35) Art. 33, the equivalent primary 
resistance is 

RJ, = I&.I + ( -*- } X 0.0062 = 3 5. 2 ohms; 

\ I2 6/ 

IiRp = 600 volts. 

which is 0.683 per cent of the primary impressed voltage. 
By Formula (47), Art. 36, when the power factor is 
unity (cos = i). 

Regulation = 0.683 +0 = 0.683 per cent 


The more correct value, as obtained from Formula (46) 
is 0.735. 

When the power factor of the load is 80 per cent, the 
approximate formula which is quite sufficiently accu- 
rate in this case gives 

Regulation = (0.683 X 0.8) + (3 . 24 X 0.6) 

= 2.5 per cent (approx.) on 80 per cent power factor. 

This is very low, and considerably less than the specified 
limit of 5 per cent. It is possible that the specified reg- 
ulation might be obtained with only 4, instead of 6, high- 
low groups of coils, and in order to produce the cheapest 
transformer to satisfy the specification, the designer 
would have to abandon this preliminary design until he 
had satisfied himself whether or not an alternative 
design with a different grouping of coils would fulfill the 
requirements. It is clear from the inspection of Fig. 52 
that an arrangement with only four L.T. coils and (say) 
sixteen H.T. coils would considerably reduce the size of 
the opening in the stampings, thus saving materials and, 
incidentally, reducing the magnetizing current, which is 
abnormally high in this preliminary design. 

Items (57) to (61). Requirements for Limiting Tem- 
perature Rise. A plan view of the assembled stampings 
should be drawn, as in Fig. 54, from which the size 
of containing tank may be obtained. In this instance 
it is seen that a tank of circular section 5 ft. 3 in. diam- 
eter will accommodate the transformer. The heiglit 
of the tank (see Fig. 55) will now have to be estimated 
in order to calculate the approximate cooling surface. 
This height will be about 90 in., and if we assume a 



smooth surface (no corrugations), the watts that can 
be dissipated continuously are 


FIG. 54. Assembled Stampings in Tank of Circular Section. 

the multiplier 0.34 being obtained from the curve, 

Fig. 32 of Art. 25. 

The watts to be carried away by the circulating 
water are (10,220+11,900) 4650= 17,470. From data 
given in Art. 29, it follows that a coil made of i| in. 



tube should have a length of 1 7>47 = 270 ft. 


r H. T. Terminal as ) 
detailed in Fig. 

FIG. 55. Sketch of isoo-k.v.a., 88,ooo-volt Transformer in Tank. 

Assuming the coil to have' an average diameter of 
4 ft. 8 in., the number of turns required will be about 25. 


On the basis of \ gal. of water per kilowatt, the 
required rate of flow for an average temperature dif- 
ference of 15 C. between outgoing and ingoing water 
is 0.25X17.47=4.37 gal. per minute. This amount 
may have to be increased unless the pipes are kept 
clean and free from scale. 

The . completed sketch, Fig. 55, indicates that a 
tank 87 in. high will accommodate the transformer 
and cooling coils, and the corrected cooling surface for 
use in temperature calculations (see Art. 25) is therefore 

18,860 sq. in. 

This new value for Item 57 has been put in the last 
column of the design sheet; but the items immediately 
following, which are dependent upon it, have not been 
corrected because the difference is of no practical im- 

Hottest Spot Temperature. The manner in which the 
temperature at the center of the coils may be calculated 
when the surface temperature is known, was explained 
in Art. 23. It is unnecessary to make the calculation 
in this instance because the coils are narrow and built 
up of flat copper strip. There will be no local " hot 
spots " if adequate ducts for oil circulation are provided 
around the coils. 

Items (62) and (63). Weight of Oil and of Complete 
Transformer. The weight of an average quality of 
transformer oil is 53 Ib. per cubic foot, from which the 
total weight of oil is found to be about 7300 Ib. The 


calculated weights of copper in the windings (Item 49) 
and iron in the core (Item 40) are 1700 Ib. and 8250 lb., 
respectively. The sum of these three figures is 17,250 lb. 
This, together with an estimated total of 4750 lb. to cover 
the tank; base and cover, cooling coil, terminals, solid 
insulation, framework, bolts, and sundries, brings the 
weight of the finished transformer up to 22,000 lb. (in- 

22 OOO 

eluding oil) ; or - - = 14.65 lb. per k.v.a. of rated full- 

load output. 

Several details of construction have not been referred 
to. It is possible, for instance, that tappings should be 
provided for adjustment of secondary voltage to com- 
pensate for loss of pressure in a long transmission line. 
These should preferably be provided in a portion of the 
winding which is always nearly at ground potential. 
It is not uncommon to provide for a total voltage varia- 
tion of 10 per cent in four or five steps, which is accom- 
plished by cutting in or out a corresponding number of 
turns, either on the primary or secondary side, which- 
ever may be the most convenient. 

Mechanical Stresses in Coils. The manner in which the 
projecting ends of flat coils in a shell- type transformer 
should be clamped together is shown in Fig. 16 of Art. 9. 
Let us calculate the approximate pressure tending to 
force the projecting portion of the secondary end coils 
outward when a dead short-circuit occurs on the trans- 
former. The force in pounds, according to Formula (4), 

JL IfL max Jj&m. 


For the quantities T and /, we have 

and /, being the average length of the portion of a 
turn projecting beyond the stampings at one end, is 

7 10.15X12 

/ = -- 27=34 in. or 86 cms. 

The value of the quantities 7 max and B &m depends on 
the impedance of the transformer. With normal full- 
load current, the impedance drop is 


where the quantities under the radical are the items 53 
and 54 of the design sheet. In order to choke back the full 
impressed voltage, the current would have to be about 

= (say) thirty times the normal full-load value. 


Thus the current value for use in Formula (4) , on the sine 
wave assumption, will be 

/max = 30 X 2 2 7 X "\/2 = 9650 amperes. 

The density of the leakage flux through the coil is less 
easily calculated; but, since the reactive voltage was 
calculated on the assumption of flux lines all parallel to 
the plane of the coil, we may now consider a path one 
square centimeter in cross-section and of length equal to 


the depth of the coil (about 29 cms.) in which the leakage 
flux will have the average value. 

-Bam = - X 2 1 X 9650 X = 4400 gausses, 

2LIO 29J 

whence, by Formula (4), 

Force in Ib. = ^ <X^5X_44o = g6 ft 


This is the force F of Fig. 16, distributed over the whole 
of the exposed surface of the end coil. An equal force 
will tend to deflect outward the secondary coil at the 
other end of the stack. If an arrangement of straps with 
two bolts is adopted as shown in Fig. 16 each bolt 
must be able to withstand a maximum load of 4370 Ib. 
Bolts f in. diameter will, therefore, be more than suf- 
ficient to prevent displacement of the coils, even on a 
dead short circuit. 



44. General Remarks, When applying the funda- 
mental principles of electrical design to special types of 
apparatus, it is necessary to consider what are the chief 
characteristics of such apparatus and wherein they differ 
from those of the more usual types. The apparatus dealt 
with in the preceding chapters is the potential transformer 
for use either, as large units, in power stations, or in 
smaller sizes, as means of distributing electric power in 
residential or industrial districts. A few special types 
of transformer will now be considered; but the treat- 
ment will be brief, with the object of avoiding useless 
repetitions. Attention will be given mainly to such dis- 
tinctive features or peculiarities as may have an impor- 
tant bearing on the design. 

45. Transformers for Large Currents and Low Volt- 
ages. Electric furnaces are built to take currents up 
to 35,000 amperes at about 80 volts usually three-phase. 
Welding transformers must give large currents at a com- 
paratively low voltage. A current of 2000 amperes at 
5 volts would probably be required for rail welding on 
an electric railroad. Transformers for thawing out 
frozen water pipes need not necessarily be specially 
designed because standard distributing transformers 



connected to give about 50 volts are used successfully 
for this purpose. A transformer of 12 k.v.a. normal 
rating, capable of giving up to 600 amperes with a max- 
imum pressure of 30 volts for short periods of time in 
cold weather, will probably answer all requirement for 
the thawing of house service pipes up to i| in. diameter. 
A current of 400 amperes will thaw out a i-in. pipe in 
about half an hour. 

In the design of all transformers for large currents, 
especially when they are liable to be practically short- 
circuited, the leakage reactance (see Art. 34) is a matter 
of importance. The permissible maximum current on a 
short circuit should be specified. In some cases, sepa- 
rate adjustable reactance coils (usually on the high- 
voltage side) are provided for the purpose of regulating 
the current from transformers used for welding and sim- 
ilar processes. 

Another point to be watched in the design of trans- 
formers for large currents is the eddy current loss in the 
copper (see Art. 20), which must be minimized by prop- 
erly arranging and laminating the secondary winding 
and leads. The mechanical details in the design of 
secondary terminals and leads also require careful atten- 

46. Constant-current Transformers. Circuits with 
incandescent or arc lamps connected in series require the 
amount of current to be approximately constant regard- 
less of the number of lamps on the circuit. If it is de- 
sired to supply series circuits of this nature from constant 
potential mains, special transformers are required, so 
designed as to give variable voltage at the secondary 


terminals, with a constant voltage across the primary 
terminals. The variations in the secondary voltage are 
automatic, being the result of very small changes in the 
secondary current, brought about by switching lamps in 
or out of the circuit. In other words, the secondary 
voltage must follow as nearly as possible the variations 
in the impedance of the external circuit, so that a doubled 
impedance would very nearly bring about a doubling of 
the secondary voltage, the drop in current being as small 
as possible. 

Automatic regulation of this kind may be obtained by 
means of an ordinary transformer having a large amount 
of magnetic leakage, as for instance a core type trans- 
former purposely constructed with the primary turns 
on one limb and the secondary turns on the other limb, 
as shown diagrammatically in Fig. i of Art. 2. The 
vector diagram of such a transformer has been drawn in 
Fig. 56, based on the simplified diagram, Fig. 48 (Art. 
36), which should be consulted for the meaning of the 
vectors. The same notation has been used in Fig. 56 
as in Fig. 48, and it is to be observed that, on account of 
the leakage flux being a large percentage of the useful 
flux, a small reduction in the current, from I\ to 7'i, will 
automatically cause the vector E e (which is a measure of 
the secondary voltage) to become E e f , just twice as great. 

Although by suitably designing a transformer with 
considerable leakage flux, a small reduction in the reactive 
drop (the vector I\X P of Fig. '56) will produce a large 
increase in the secondary voltage, it is obvious that still 
better results would be obtained if the reactance (or 
amount of leakage flux) could be made to decrease at a 



greater rate than the current. Thus, if an increase of 
current could be made to bring about a change in the 
permeance of the leakage paths, the reactive drop, 
instead of being proportional to the current, might be 
made to increase at a greater rate than the current, 

FIG. 56. Vector Diagram of Transformer with Large Amount of Leak- 
age Flux. 

and so bring about the condition illustrated by Fig. 57 
where the same result, i.e., a doubling of the secondary 
voltage is seen to be brought about by a very much 
smaller reduction in the amount of the current. 

It is evident that the primary volt-amperes must 
remain practically constant at all loads, and the fact that 


the actual secondary output may vary considerably 
with changes in the resistance of the external circuit, is 
accounted for by the alteration in the power factor of 
the primary circuit. Thus, since the input and output 
of a transformer must be the same except for the internal 
losses, the changes of input with an almost constant 


FIG. 57. Vector Diagram of Transformer with Variable Leakage 

Ej>Ip product are accounted for by the changes in the 
angle </> of Fig. 57. 

Fig. 58 illustrates the principle of construction of the 
constant-current transformer with variable magnetic 
leakage. One coil is stationary while the other is movable, 
being suspended from a pivoted arm provided with a 
counterweight, and free to slide up and down on the cen- 



tral core of a shell-type magnetic circuit. The movable 
coil may be either the primary or the secondary, and by 
careful adjustment of the balance weight, a very small 
jchange in the current may be made to produce a con- 
siderable change in the relative position of the coils, thus 
greatly altering the relation between the- leakage and 

FIG. 58. Constant Current Transformer with " Floating " Coil. 

useful flux components, the (vectorial) sum of which 
passing through the primary coil must always remain 
practically constant. 

With the two coils in contact, the maximum secondary 
voltage corresponding to the maximum number of 
lamps in series is obtained; while on short-circuit the 
movable coil will be pushed as far away from the sta- 


tionary coil as the construction of the transformer will 
admit. Except for the difficulty of calculating accu- 
rately the amount of the leakage flux-linkages corre- 
sponding to these two conditions, the design of a 
constant-current transformer for any given output is 
a simple matter. Regulation is not usually required 
over a range greater than from full load to about one- 
third of full load, and this can be obtained with a cur- 
rent variation not exceeding i per cent. 

The force tending to move the coils apart can readily 
be calculated with the aid of Formula (4) Art. 9; but 
since the quantity B am cannot be predetermined with 
great accuracy except in the case of standard designs 
for which data have been accumulated final adjustments 
must be made after completion, by the proper setting of 
the counterweight. 

Constant-current transformers for arc-lamp circuits 
off constant pressure mains require a secondary current 
between 6.5 and 10 amperes, and they usually operate 
in conjunction with a mercury arc rectifier to change the 
alternating current into a continuous current. Trans- 
formers for small outputs may be air cooled, while the 
larger units should be oil-immersed and, if necessary, 
cooled by circulating water. 

The full-load efficiency of constant-current trans- 
formers with movable coils for use on 2200-volt circuits 
ranges from 90 per cent for 3 kw. output on 6o-cycle 
circuits to 96 per cent for 30 kw. output on 25-cycle 

47. Current Transformers for Use with Measuring 
Instruments. These transformers are of comparatively 


small size, their chief function being to provide a current 
for measuring-instruments which shall be as nearly as 
possible proportional to the line current passing through 
the primary coils. By their use it is possible to trans- 
form very large currents to a current of a few amperes 
which may conveniently be carried to instruments of 
standard construction mounted on the switchboard 
panels or in any convenient position preferably not very 
far removed from the primary circuit. Again, in the 
case of high-potential circuits, even if the reduction of 
current is not of great importance, the fact that the sec- 
ondary circuit of the current transformer can be at 
ground potential renders unnecessary the special instru- 
ments and costly methods of insulation that would be 
required if the line current of high- voltage systems were 
taken through the measuring instruments. 

A current transformer does not differ fundamentally 
from a potential transformer; but since the primary 
coils are in series with the primary circuit, the voltage 
across the terminals will depend upon the induced 
volts, which, in their turn, depend upon the impe- 
dance of the secondary circuit. With the secondary 
short-circuited, the voltage absorbed will be a mini- 
mum, and the input of the transformer will be approxi- 
mately equal to the copper losses, because a very small 
amount of flux will then be sufficient to generate the 
required voltage, and the iron losses will be negligible. 

The vector diagram for a series transformer does not 
differ from that of a shunt transformer, but Figs. 59 
and 60 have been drawn to show clearly the influence 
of the magnetizing current on the relation between 



the total primary and secondary currents. Fig. 59 
shows the- vector relations when the power factor is 
unity, while in Fig. 60 there is an appreciable lag be- 
tween the current and e'.m.f. in the secondary circuit. 

When a current transformer is used in connection 
with an ammeter only, the essential condition to be 

fulfilled is that the ratio 

/./ 1 1 

7\ or T 

* p \ *p 

be as nearly 

constant as possible over the whole range of current 
values. When the secondary current is passed through 


FIG. 59. 

the series coil of a wattmeter, it is equally important 
that L be as nearly as possible opposite in phase to 
IP, or, in other words, that the angle I P OIi be very small. 

A diagram, such as Fig. 60, may be constructed for 
any given condition of load, the amount of the flux 
B and therefore the exciting current I e being de- 
pendent upon the impedance of the secondary circuit, 
since this determines the necessary secondary voltage. 

On the sine wave assumption, it is an easy matter 
to express the quantity 01 p in terms of the secondary 


current and the two components of the exciting current. 
The vector 01 \ is a measure of the secondary current, 

IT \ 
being simply I A - ), and it is easily seen that 

M vl 

IP = \/(/i sin 0+/ ) 2 + (/i cos d+I u y, 

whence the ratio can be calculated for any power 

1 v 

factor (cos 0) and any values of the secondary current 

D C 

FIG. 60. 

and voltage. It is interesting to note that, on a load 
of unity power factor (cos 0=i), the magnetizing com- 
ponent of the total exciting current does not appre- 
ciably affect the relation between the magnitudes of 
the primary and secondary currents, and for all practical 


purposes the difference, under this particular load con- 
dition, is 

,.,.,. iron loss (watts) 

L-D /I =-/,= 

e.m.f . induced in primary (volts) ' 

If this difference were always proportional to the 
primary current, there would be no particular advantage 
in keeping it very small; but since the power factor 
is not always unity, and variations in current mag- 
nitudes may be brought about by phase differences, 
it is always advisable to aim at obtaining an exciting 
current which shall be a very small percentage of the 
total primary current. 

The phase difference between I p and I\ (see Figs. 
59 and 60) may be expressed as 

This angle must be very small, especially when the 
transformer is for use with a wattmeter. It should 
never exceed i minute, and should preferably be within 
thirty seconds. This condition can only be satisfied, 
with varying values of 6, by making the exciting current 
(especially the magnetizing component 7 ) very small 
relatively to the main current. It is therefore neces- 
sary to use low flux densities in ^the cores of series trans- 
formers for use with instruments, and this incidentally 
leads to small core losses and a small " energy " com- 
ponent (Iw) of the total exciting current. Flux densities 
ranging from 1500 to 2500 gausses at full load are not 


uncommon in well-designed series instrument trans- 
formers. Fig. 6 1 gives approximate losses per pound 
of transformer iron for these low densities which are 
not included in the curves of Fig. 27. Although curves 
for alloyed steel are not given, the losses may be 
approximately estimated by referring to Fig. 27 (Art. 
20) and noting the relative positions of the curves for 
the two qualities of material. 

When the primary current is large, a convenient 
form of current transformer is one with a single turn of 
primary, that is to say, a straight bar or cable passing 
through the opening in the iron core. This is quite 
satisfactory for currents of 1000 amperes and upward, 
and the construction is t permissible with currents as 
low as 300 amperes, especially when the transformer 
is to be used 'in connection with a single ammeter, i.e., 
without a wattmeter, or second instrument, or relay 
coil, in series. The designer should, however, aim to 
get 1000 to 1500 ampere turns, or more, in each winding 
of a series instrument transformer. 

Although the presence of the exciting current com- 
ponent of an iron-cored transformer renders a constant 
ratio of current transformation theoretically unattain- 
able over the whole range of current values, this does 
not mean that any desired ratio of transformation 
cannot be obtained for a particular value of the primary 
current. It is, of* course, a simple matter to eliminate 
the error due to the presence of the exciting current 

(T \ 
-] that any 

desired current transformation may be obtained for a 





0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

Total watts per pound sw 

FIG. 61. Losses in Transformer Iron at Low Flux Densities. 



specified value of the primary current. If the ratio of 
transformation is correct at full load, it will be prac- 
tically correct over the range from f to full-load cur- 
rent, the error being most noticeable with the smaller 
values of the main current. The following figures are 
typical of the manner in which the transformation 
ratio of series instrument transformers is likely to vary. 

Percentage of Full-load 







I .0 



o. 16 

i .0 









Column A gives average values: column B shows 
how small the error may be in well-designed transformers 
for use with wattmeters or other instruments demand- 
ing constancy in the current ratios; while C refers to 
commercial current transformers for use with relays, 
trip coils of switches, and other apparatus which does 
not call for great accuracy in the transforming ratio. 
In all cases a fairly low power factor is assumed, and a 
rated full-load output of about 50 volt-amperes. If 
the same transformers were to operate on an external 
circuit of reduced resistance and unity power factor, 
the percentage error would be considerably smaller. 

No special features other than reliability of insulation, 
and freedom from overheating have to be considered 



in connection with series transformers used for oper- 
ating regulating devices or protective apparatus such 
as trip coils on automatic overload circuit-breakers. 
The flux density in the core may then be higher than 
in instrument transformers. 

48. Auto-transformers. An ordinary transformer be- 
comes an auto-transformer, or compensator, when the 

FIG. 62. Ordinary Transformer Connected as Auto-transformer. 

connections are made as in Fig. 62. One terminal is 
then common to both circuits, the supply voltage 
being across all the turns of both windings in series, 
while the secondary or load voltage is taken off a por- 
tion only of the total number of turns. This arrange- 
ment would be adopted for stepping down the voltage; 
but by interchanging the connections from the supply 


circuit and the load, the auto- transformer can be used 
equally well for stepping up the voltage. 

There is little advantage to be gained by using auto- 
transformers when the ratio of transformation is large; 
but for small percentage differences between the supply 
and load voltages, considerable economy is effected by 
using a,n auto-transformer in place of the usual type 
with two distinct windings. 

Let 7\, = the number of turns between terminals 

a and c (Fig. 62) ; 
T s = the number of turns between terminals 

c and b; 

then (T p +Ts) = ihe number of turns between terminals 
a and b. 

The meaning of other symbols is indicated on Fig. 62. 
The ratio of transformation is 

E v p s ( x 

= - =r (54) 

If used as an ordinary transformer, the transforming 
ratio would be 

-r-i (55) 

J- s 

The ratio of currents is 

-^-r, (56) 


while the current I c in the portion of the winding com- 
mon to both primary and secondary is obtained from 
the equation 

1 cJ- s 1 pi p, 


/ = /,('-!), ..... (57) 

or, in terms of the secondary current, 


None of the above expressions takes account of the 
exciting current and internal losses. 

The volt-ampere output, as an auto-transformer, is 
E s ls'y but part of the energy passes directly from the 
primary into the secondary circuit. For the purpose 
of determining the size of an auto-transformer, we 
require to know its equivalent transformer rating. 
The volt-amperes actually transformed are E S I C , whence 

Output as ordinary transformer _Ic_r i . ^ 
Output as auto- transformer I s r 

which shows clearly that it is only when the ratio of 
voltage transformation (r) is small that an appreciable 
saving in cost can be effected by using an auto-trans- 

The ratio of turns, and the amount of the currents 
to be carried by the two portions of the winding having 
been determined by means of the preceding formulas, 
the design may be carried out exactly as for an ordi- 



nary potential transformer, attention being paid to 
the voltage to ground, which may not be the same in 
the auto-transformer as in an ordinary transformer for 
use under the same conditions. Auto-transformers are, 
however, rarely used on high voltage circuits, although 
there appears to be no objection to their use on grounded 

Effect of the Exciting Current in Auto-transformers. 
In the foregoing discussions, the effect of the exciting 
current was considered negligible. Tin's assumption is 



Tp turns Tg turns 







FIG. 63. Diagram of Connections of Auto- transformer. 

usually permissible in practice; but since it may some- 
times be necessary to investigate the effect of the 
exciting current components, a means of drawing the 
vector diagram showing the correct relation of the 
current components will now be explained. 

Fig. 63 is similar to Fig. 62 except that it shows 
the connections in a simplified manner. The arrows 
indicate what we shall consider the positive directions 
of the various currents. 


The fundamental condition to be satisfied is that the 
(vectorial) addition of all currents flowing to or from 
the junction c or b shall be zero. Whence, 

Ip + Is = Ic ...... (60) 

Let L stand for the exciting current when there is 
no current flowing in the secondary circuit. This is 
readily calculated exactly as for an ordinary trans- 
former with Ep volts across (T P + T S ) turns of winding. 
Then, since the resultant exciting ampere turns must 
always be approximately (Tp-\-T s )I e , the condition to 
be satisfied under load is 

V, (6i) 
which, if we divide by T s , becomes 

If I c in this equation is replaced by its equivalent 
value in terms of the other current components, as 
given by Equation (60), we get j s 

-- / f 

rl p = rr e -l s (63) 

The vector diagram Fig. 64 satisfies these conditions; 
the construction being as follows: 

Draw OB and OE S to represent respectively the phase 
of the magnetic flux and induced voltage. Draw OI 
to represent the current in the secondary circuit in 
its proper phase relation to E s . Now calculate the 



exciting current I e on the assumption that it flows 
through all the turns (T P +T S ], and draw OM, equal 
to rI C j in its proper phase relation to OB. Join ML 

and determine the point C by making CI S = - . Then, 

since I S M is the vectorial difference between rl e and 
I s , it follows from Equation (63) that it is equal to 


FIG. 64. Vector Diagram of Auto-transformer, Taking Account of 
Exciting Current. 

rI P , whence CI S =I V , and CM = (ri)I p . Also, since 
OC is the vectorial sum of I s and I p , it follows from 
Equation (60) that OC is the vector of the current I c 
in the portion of the winding common to both circuits. 
In this manner the correct value and phase relations 
of the currents ID and 7 C , in the sections ac and cb of 
the winding, can be calculated for any given load 


49. Induction Regulators. In order to obtain a vari- 
able ratio of voltage transformation, it is necessary either 
to alter the ratio of turns by cutting in or out sections 
of one of the windings, or to alter the effective flux- 
linkages by causing more or less of the total flux linking 
with the primary to link with the secondary. 

The principle of variable ratio transformers of the 
moving iron type is illustrated by the section shown 
in Fig. 65. This is a diagrammatic representation of 
a single-phase induction regulator with the primary 
coils on a cylindrical iron core capable of rotation 
through an angle of go degrees. The secondary coils 
are in slots in the stationary portion of the iron cir- 
cuit. The dotted lines show the general direction of 
the magnetic flux when the primary is in the position 
corresponding to maximum secondary voltage. As the 
movable core is rotated either to the right or left, the 
secondary voltage will decrease until, when the axis 
AB occupies the position CD, the flux lines linking 
with the secondary generate equal but opposite e.m.f.s 
in symmetrically placed secondary coils, with the result 
that the secondary terminal voltage falls to zero. If 
current is flowing through the secondary winding 
as will be the case when the transformer is connected 
up as a " booster " or feeder regulator the reactive 
voltage due to flux lines set up by the secondary current 
and passing through the movable core in the general 
direction CD, will be considerable unless a short-circuited 
winding of about the same cross-section as these cond- 
ary is provided as indicated in Fig. 65. 

It is immaterial whether the winding on the movable 



core be the primary or secondary ; but if the primary is 
on the stationary ring, the short-circuited coils must 
also be on the ring. 

The chief difficulty in the design of induction regulators 

FIG. 65. Diagram of Single-phase Variable-ratio Transformer of the 
Moving-iron Type. 

arises from the introduction of necessary clearance 
gaps in the magnetic circuit, and the impossibility of 
arranging the coils as satisfactorily as in an ordinary 
static transformer so as to avoid excessive magnetic 


leakage. A large exciting current component and an 
appreciable reactive voltage drop are characteristic of 
the induction voltage-regulator. 

Fig. 66 is a diagram showing a single-phase regu- 
lating transformer of the type illustrated in Fig. 65 
connected as a feeder regulator, the secondary being 
in series with one of the cables leaving a generating station 
to supply an outlying district. The movement of the 
iron core can be accomplished either by hand, or auto- 
matically by means of a small motor which is made to 
rotate in either direction through a simple device 
actuated by potential coils or relays. 

The lower diagram of Fig. 66 shows the core carrying 
the primary winding in the position which brings the 
voltage generated in the ring winding to zero. The 
flux lines shown in the diagram are those produced 
by the magnetizing current in the primary winding; 
but there are other flux lines not shown in the diagram 
which are due to the current in the ring winding. 
It is true that the movable core carries a short-cir- 
cuited winding not shown in Fig. 66 which greatly 
reduces the amount of this secondary leakage flux; 
but it will nevertheless be considerable, and the secondary 
reactive voltage drop is likely to be excessive, especially 
if the ring winding consists of a large number of turns. 
An improvement suggested by the writer at the time * 
when this type of apparatus was in the early stages of 
its development, consists in putting approximately half 
the secondary winding on the portion of the magnetic 
circuit which carries the primary winding, the balance 

* The year 1895. 


FIG- 66. Variable-ratio Transformer Connected as Feeder Regulator. 


of the secondary turns being put on the other portion 
of the magnetic circuit. The connections are made as in 
Fig. 67, the result being that the movement of the 
rotating core, to produce the full range of secondary 
voltage from zero to the desired maximum, is now 180 
instead of 90 as in Fig. 66; but since, under the same 
conditions of operation, the ring winding for a given 
section of iron will carry only half the number of turns 
that would be necessary with the ordinary type (Fig. 
66), the secondary reactive voltage drop is very nearly 
halved. This is one of the special features of the 
regulating transformers manufactured by Messrs. 
Switchgear & Cowans, Ltd., of Manchester, England. 
Consider the case of a single-phase system with 2200 
volts on the bus bars in the generating station. The 
voltage drop in a long outgoing feeder may be such as 
to require the addition of 200 volts at full load in order 
to maintain the proper pressure at the distant end. If 
this feeder carries 100 amperes at full load, the neces- 
sary capacity of a boosting transformer of the type 
shown diagrammatically in Fig. 67 is 20 k.v.a. This 
variable-ratio transformer, with its primary across the 
2 200- volt supply, and its secondary in series with the 
outgoing feeder, will be capable of adding any voltage 
between o and 200 to the bus-bar voltage. As an 
alternative, the supply voltage at the generating station 
end of this feeder may be permanently raised to 2300 
volts by providing a fixed-ratio static transformer 
external to the variable-ratio induction regulator and 
connected with its secondary in series with the feeder. 
An induction regulator of the ordinary type (Fig. 66) 



Position of Zero 
Secondary Pressure 

Position of Maximum 
Secondary Pressure 

FIG. 67. Moving-iron Type of Feeder Regulator with Specially Drranged 
Secondary Winding. 


capable of both increasing and decreasing the pressure 
by 100 volts, will then provide the desired regulation 
between 2200 and 2400 volts. The equivalent trans- 

former output of this regulator will be = 10 



The Polyphase Induction Regulator. Two or three 
single-phase regulators of the type illustrated in Fig. 65 
may be used for the regulation, of three-phase circuits; 
but a three-phase regulator is generally preferable. The 
three-phase regulator of the inductor type is essentially 
a polyphase motor with coil-wound not squirrel-cage 
rotor, which is not free, to rotate, but can be moved 
through the required angle by mechanical gearing oper- 
ated in the same manner as the single-phase regulator. 
The rotating field due to the currents in the stator coils 
induces in the rotor coils e.m.f.'s of which the magnitude 
is constant, since it depends upon the ratio of turns, 
but of which the phase relation to the primary e.m.f. 
depends upon the position of the rotor coils relatively 
to the stator coils. When connected as a voltage 
regulator for a three-phase feeder, the vectorial sum of 
the secondary and primary volts of a three-phase 
induction regulator will depend upon the angular dis- 
placement of the secondary coils relatively to the cor- 
responding primary coils. 

Mr. G. H. Eardley-Wilmot * has pointed out certain 

advantages resulting from the use of two three-phase 

induction regulators with secondaries connected in series, 

for the regulation of a three-phase feeder. By making 

* The Electrician, Feb. 19, 1915, Vol. 74, page 660, 


the connections so that the magnetic fields in the two 
regulators rotate in opposite directions, the resultant 
secondary voltage will be in phase with the primary 
voltage. The torque of one regulator can be made to 
balance that of the other, thus greatly reducing the 
power necessary to operate the controlling mechanism. 




Absolute unit of current 26 

Air-blast, cooling by, 88 

All-day efficiency (see Efficiency). 

Alloyed-iron transformer stampings 19 

Ampere-turns to overcome reluctance of joints 127 

Analogy between dielectric, and magnetic, circuits 33 

Auto-transformers 191 


B-H curves (see Magnetization curves). 

Bracing transformer coils (see Stresses in transformer coils). 

Bushings (see Terminals). 

Calorie, definition 99 

Capacity current 41 

electrostatic 33, 36 

of plate condenser 40 

Capacities in series 42 

Charging current (Capacity . current) 41 

Classification of transformers 14 

Compensators 191 

Condensers in series 42 

Condenser type of bushing , 62 

Conductivity, heat 80, 82, 87 

Constant-current transformers 178 

Construction of transformers 17, 24, 31 


206 INDEX 


Cooling of transformers 14, 88, 91, 103 

by air blast 88 

forced oil circulation 106 

water circulation 105 

Copper losses 72, 75, 76, 83, 142, 165 

resistivity of 144 

space factor (see Winding space factor). 

Core loss (usual values) (sec also Losses in iron) 77 

Core-type transformers 17, 22 

Corrugations, effect of, on sides of tank 94 

on insulator surface 60 

Coulomb 34 

Current density in windings 72 

transformers 184 


Density (see Flux- and Current-density). 

Design coefficient (c) 149 

numerical example in 154 

problems 13 

procedure in : 150 

Dielectric circuit 32 

constant 36 

constants, table of 37 

strengths, table of 37 

Disruptive gradient 36, 62 

Distributing transformers 17 


Eddy currents in copper windings 73 

current losses (see Losses). 

Effective cooling surface of tanks 96 

Efficiency. . . '. 73, 167 

all-day 74 

approximate, of commercial transformers 74, 183 

calculation of, for any power factor 77 

maximum 145 

Elastance, definition 35 

INDEX 207 


Electrifying force 38 

Electrostatic force 38 

E.m.f. in transformer coils (see also Volts; Voltage) 4, 5, 6 

Equivalent cooling surface of tanks 96 

ohmic voltage drop. 134, 137 

Exciting current 5, 125, 168 

in auto-transformers 194 

volt-amperes 129 

(curves) 131 


Farad 33 

Flux density, electrostatic 35 

in transformer cores i. 72, 164 

leakage (see Leakage flux). . 

Forces acting on transformer coils 24, 1 74 

Frequency, effect of, on choice of iron . . 19 

allowance for core loss 77 

Furnaces, transformers for electric 177 


Heat conductivity of materials 80 

copper 83, 87 

insulation 87 

Heating of transformers (see Temperature rise). 

High-voltage testing transformers 15 

Hottest spot calculations 84 

Hysteresis, losses due to (see Losses). 


Induction regulator 197 

polyphase 203 

Instrument transformers 183 

Insulation of end turns of transformer windings 50 

oil 52 

problems of transformer 32 

thickness of 48 

Iron, losses in 69, 77, 142, i8c 

208 INDEX 



Laminations, losses in 69, 77, 142, 189 

shape of, in shell-type transformer 19 

thickness of 19 

Large transformers. . 16, 17 

Leakage flux : 98, 107, 118, 179, 198 

reactance (see Reactance; Reactive voltage drop). 

Losses, eddy current 69 

hysteresis 69 

in copper windings 72, 75, 76, 83, 142, 165 

in iron circuit 69, 77, 142, 189 

power, in transformers 69 


Magnetic leakage (see Leakage flux). 

Magnetization curves for transformer iron % 128 

Magnetizing current (see Exciting current). 

Mechanical stresses in transformers 24, 1 74 

Microfarad 36 


Oil insulation. , 52 

Output equation 138 

Overloads, effect of, on temperature 98 


Permeance 34, 39 

Permittance (see Capacity). 

Polyphase transformers 12, 22 

Potential gradient 38 

Power losses (see Losses). 

transformers 16, 154 


Quantity of electricity (Coulomb) 34 

INDEX 209 



Reactance, leakage, experimental determination of 114 

Reactive voltage drop 117, 137, 180 

Regulation 109, 132, 168 

formulas 134, 135 

Regulating tranformers 197 

polyphase 203 

Reluctance, magnetic 35 

Resistance of windings 165 

thermal 81 

Resistivity of copper 144 


"Sandwiched" coils 118 

Saturation; reasons for avoiding high flux densities 164 

Self-induction of secondary winding 108 

Series, transformers 184 

Shell-type transformers 17, 20, 24, 155 

Short-circuited transformer, diagram of 116 

Silicon-steel for transformer stampings 71 

Single-phase units used for three-phase circuits 12 

Space factor, copper (see Winding space factor) . 

iron 151 

Sparking distance; in air 58, 68 

in oil 52, 53 

Specifications ' 140, 155 

Specific inductive capacity (see Dielectric constant). 

heat; of copper 99 

of oil 99 

Stacking factor 151 

Stampings, transformer, thickness of 19 

Static shield on h.t. terminals 65, 68 

Stresses in transformer coils 24, 1 74 

Surface leakage 46 

under oil 54 

Symbols, list of.' ix 

210 INDEX 



Temperature rise of transformers 79, go, 92, 94, 98, 170 

after overload of short duration 99 

Terminals ; 54 

composition-filled 59 

condenser type 62 

oil-filled 57, 60 

porcelain 57 

Test voltages. 58 

Theory of transformer, elementary 2 

Thermal conductivity (see Heat conductivity). 

ohm, definition 81 

Three-phase transformers 12, 22 

Transformers, auto 191 

constant current 178 

core-type 17, 20, 22 

current 184 

distributing 17 

for electric furnaces 177 

large currents 178 

use with measuring instruments 183 

polyphase 12, 22 

power 16, 154 

series : . . . . 184 

shell-type 17, 20, 24, 155 

welding 177 

Tubular type of transformer tank 104 


Variable-ratio transformers. 197 

Vector diagram illustrating effect of leakage flux no, 112 

of auto-transformer 196' 

short-circuited transformer 1 16 

series transformer 185, 186 

transformer on inductive load n, i$$. 134, 135 

non-inductive load . . .' 10 

INDEX 211 


Vector diagram of transformer with large amount of leakage flux. . . 180 

open secondary circuit 5 

variable leakage reactance 181 

showing components of exciting current 126 

Voltage, effect of, on design 15 

drop due to leakage flux 117, 137 

regulation (see Regulation). 

Voits per turn of winding < 141 


Water-cooled transformers 105 

Weight of transformers 151, 173 

Welding transformers 177 

Windings, estimate of number of turns in 141 

Winding space factor 51, 151, 152 

:t Window," dimensions of, in shell-type transformers 160, 163 

Wire, size of, in windings (see Current density). 




Air clearances (Formula) 49 

quantity required for air-blast cooling 89, 90 

Ampere-turns, allowance for joints 127 


B-H curves (Gausses and amp- turns per inch) 128 


Capacity current 42 

in terms of dimensions, etc 36 

Charging current 42 

Cooling area of tanks (Curve) 93 

Copper space factors 51, 151, 152 

Core loss (usual values) 77 

Corrugated tanks, correction factor for cooling surface of 96 

Current density (usual values) 72 


Density, current, in coils (usual values) 72 

in transformer cores (Table) 72 

Dielectric constants (Table) 37 

strengths (Table) 37 

Disruptive gradient (see Dielectric strength). 


Efficiency (usual values) 74 

E.m.f., formulas 5, 6 



Equivalent surface of corrugated tanks (correction factor) 96 

Exciting volt-amperes, Formula 130 

Curve '. 131 

Flux densities in core (Table) 72 

Force exerted on coil by leakage flux 28 


Hottest spot tempe.-ature -(Formula) 86 

Inductive voltage drop (Formula) 124 

Insulation, air clearance 49 

oil clearance . 53 

thickness of (Table) 48 

Iron loss (Curves) 70, 189 


Joints in iron circuit, ampere turns required for 127 

Losses in cores (usual values) 77 

transformer iron (Curves) 70, 189 


Magnetization curves for transformer iron 128 

Magnetizing volt-amperes (Curve) 131 

Mechanical force on coil due to magnetic field 28 


Oil, insulation thickness in 53, 54 

transformer, test voltages 52 

Output equation , 138 




Power losses in transformer iron (Curves) 70, 189 


Reactance, leakage, in terms of test data 117 

Reactive voltage drop (Formula) 124 

Regulation formulas 135, 136 

Resistance, equivalent primary 117 


Space factors, copper 51, 151, 152 

iron 151 

Specific inductive capacity (Dielectric constant), (Table) ......... 37 

Surface leakage distance, in air 50 

under oil.. 54 


Temperature of hottest spot (Formula) 86 

rise due to overloads (Formula) 98, 101 

in terms of tank area (Curve) 93 

Thickness of insulation 48 

in oil 53, 54 


Voltage drop, reactive (Formula.) 124 

regulation (Formulas) 135, 136 

Volt-amperes of excitation, (P'ormula) 130 

(Curves) 131 

Volts per turn of winding (Formula) '. . 142 

numerical constants 149 


Water, amount of, required for water-cooling coils 105 

Winding space factors 51, 151, 152 




Capacities in series 43 

Composition-filled bushing 58 

Condenser-type bushing 65 

Cooling-coil for water-cooled transformers 105, 171 

"Hottest spot" temperature calculation 87 

Layers of different insulation in series 44 

Mechanical stresses in coils 1 74 

Plate condenser 41 

Temperature rise due to overloads 98, 102 

of self cooling oil-immersed transformer 94 

with tank having corrugated sides 97 

Transformer design 154 

Voltage regulation 136, 168 

Volt-amperes of excitation per pound of iron in core 130 







i o 


MAR 2 6 1948 

AUG 27 1948 



AU6 22 1949 

\ : 0y 


W V2 5 1949